This is a controversial subject both historically and politically. It is not settled history and is debated by historians on both sides of the issue. Given that here is the case made for genocide.

Genocide is defined as the deliberate and systematic extermination of a national, racial, political, or cultural group. By that basis I've seen it argued the British did not commit genocide because some of the British policies which created the famine predate the famine by decades and even centuries. I've seen it argued, incompetence and indifference, does not rise to the level of "deliberate". The argument in favor of genocide makes note that Ireland is 100 miles off the coast of the UK, and to dismiss Whitehall's inaction to incompetence given a third of the population of Ireland slowly starved to death on their doorstep over seven years is refuted by Whitehall exhibited no such incompetence when reacting to famines in other parts of the UK (such as the Scottish famine caused by the same potato blight and which occurred at the same time). Also, other than the response to the Great Irish Famine, these same politicians are not recognized for their incompetence so much as their stern policies towards Ireland.

The British in the mid 1800's were certainly indifferent to the Irish Famine.

Charles Trevelyan, who was in charge of the administration of government relief, limited the Government's food aid program because of a firm belief in laissez-faire. He thought that "the judgement of God sent the calamity to teach the Irish a lesson".
...
whilst in Scotland he (Trevelyan) was closely associated with the work of the Central Board for Highland Relief. His inaction and personal negative attitude towards the Irish people are widely believed to have slowed relief for the famine.

Trevelyan placed charge of the British response, or failure to respond under the Whig Party's Lord Russell. Lord Russell came to power when the previous Prime Minister Sir Robert Peel tried to repeal the Corn Laws to help relieve the Irish Famine. Peel tried twice to accomplish this, the second time his government coalition fractured and he was ushered out of power. Lord Russell therefore came to power with a mandate to take a harder line on the Irish.

Indifference exhibited by a lack of response wasn't Britain's only contribution to creating the famine.

IN 1997 Tony Blair, the British prime minister, made the first formal apology for Britain’s role in the Irish famine.

Perhaps what Blair was apologizing for included a series of British laws designed to punish and weaken the Irish ruling and middle classes which left the country economically underdeveloped, uneducated, and unskilled. The centuries of systemic discrimination under British law left the Irish especially vulnerable in 1840 to address the famine on their own. Many of the Irish due these coercive laws, were relegated to subsistence farming on small parcels of land where few crops could be grown which would yield the number of calories required to support the tenants. The potato was one such crop and became a staple for the Irish. Thus, when the potato blight hit globally, the accompanying famine only hit the poorest, most vulnerable, who had been left systemically unprotected; the Irish. On the European Continent, United States, and elsewhere, which were also effected by the potato blight, other crops softened the impact of the absence of the potato crop. Thus the effects of the blight did not result in famine.

Ireland too had other crops, but here too British policy dissuaded these crops from being used to ease the famine. Crops exported from Ireland were taxed at a lower rate, so they could be sold in Britain for a lucrative price. Crops from elsewhere in the world brought into Ireland were taxed at the higher rate so they were prohibitively expensive to procure. While the population of Ireland was starving, Ireland exported much of there food through out the famine. There were also cases where the British government interceded to discourage foreign governments from giving aid to the Irish during the famine.

From a historical perspective, The Great Famine(1845–1852) was not an isolated event. It was actually the second of three famines which hit Ireland. The first famine or Irish Famine also under British Rule occurred in (1740–41) and killed between 20-40% of the Irish people. The Last famine occurred 1879 was not as deadly as the first two; however it did fuel Irish immigration to the United States.

Lastly, there is a history of famine being used as a marshal/political tool in conflicts between the Irish and the British. In the 9 years war (1593 to 1603) fought between Gaelic Irish chieftains and the English Crown, 100,000 people mostly civilians died on the Irish side, the vast majority due to starvation. Starvation intentionally caused by the British which targeted the Civilians in the conflict.

9 years war
The English scorched earth tactics were especially harsh on the civilian population, who died in great numbers both from direct targeting and from famine.

Likewise when Cromwell conquered Ireland 1649–53, 20-40% of the civilian population of Ireland died, due to both famine created as a strategy to win the war and by direct targeting.

Cromwell conquered Ireland John Hewson systematically destroyed food stocks in counties Wicklow and Kildare, Hardress Waller did likewise in the Burren in County Clare, as did Colonel Cook in County Wexford. The result was famine throughout much of Ireland, aggravated by an outbreak of bubonic plague.[26] As the guerrilla war ground on, the Parliamentarians, as of April 1651, designated areas such as County Wicklow and much of the south of the country as what would now be called free-fire zones, where anyone found would be, "taken slain and destroyed as enemies and their cattle and good shall be taken or spoiled as the goods of enemies". This tactic had succeeded in the Nine Years' War.

One persuasive counter argument to these events is they occurred two hundred years removed from the Great Irish Famine and occurred in a different time and different age. The rebuttal to that is the punitive laws designed to impoverish the Irish adopted after these rebellions, remained in effect for hundreds of years or nearly right up until the Great Irish Famine occurred. This it's argued indicates the relationship between the peoples hadn't advance much in those interceding centuries.

That's basically the case. The Great Irish Famine wasn't an isolated event under British rule. Famine was used as a tool against the Irish historically. British Laws which severely impacted economic, educational and political opportunity; and created the vulnerability in the Irish people that the potato blight exploited to create the famine. British politicians who had the authority and means to intervene chose not to. Their stated reasons included ones based upon race and religion. The person in charge of the British relief(Trevelyan) went so far as to equate the famine with God's judgement on the Irish. Finally far from easing the effects of the Great Famine the British policy makers continuously made the ongoing crisis worse by limiting food imports, promoting food exports, and discouraging foreign aid. The combination of these policies would create a famine which would consume a third of the people of Ireland where the rest of the world subject to the same potato blight did not experience any such phenomena.

Detailed Answer

The Irish Great Famine occurred from 1845–1852.

- A series of laws designed to punish and weaken the Irish ruling and middle class left the country uneducated, unskilled and vulnerable.

These laws included the Act of Uniformity, the Test Acts and the penal laws.

Penal Laws

• Popery Act – Catholic inheritances of land were to be equally subdivided between all an owner's sons with the exception that if the eldest son and heir converted to Protestantism that he would become the one and only tenant of estate and portions for other children not to exceed one third of the estate. This "Gavelkind" system had previously been abolished by 1600. [1703-1829); This turn large estates into small subsistence farms.
• 'No person of the popish religion shall publicly or in private houses teach school, or instruct youth in learning within this realm' upon pain of twenty pounds fine and three months in prison for every such offense. Repealed in 1782.
• Exclusion of Catholics from most public offices (since 1607)
• Presbyterians were also barred from public office from 1707
• Ban on intermarriage with Protestants; repealed 1778
• Presbyterian marriages were not legally recognized by the state
• Catholics barred from holding firearms or serving in the armed forces (rescinded by Militia Act of 1793)
• Bar from membership in either the Parliament of Ireland or the Parliament of England from 1652; rescinded 1662–1691; renewed 1691–1829, applying to the successive parliaments of England (to 1707), Great Britain (1707 to 1800), and the United Kingdom of Great Britain and Ireland (1800 to 1829).
• Disenfranchising Act 1728, exclusion from voting until 1793; Exclusion from the legal professions and the judiciary; repealed (respectively) 1793 and 1829.
• Education Act 1695 – ban on foreign education; repealed 1782. Bar to Catholics and Protestant Dissenters entering Trinity College Dublin; repealed 1793.
• On a death by a Catholic, his legatee could benefit by conversion to the Church of Ireland;
• Ban on converting from Protestantism to Roman Catholicism on pain of Praemunire: forfeiting all property estates and legacy to the monarch of the time and remaining in prison at the monarch's pleasure. In addition, forfeiting the monarch's protection. No injury however atrocious could have any action brought against it or any reparation for such. Ban on Catholics buying land under a lease of more than 31 years; repealed 1778.
• Ban on custody of orphans being granted to Catholics on pain of 500 pounds that was to be donated to the Blue Coat hospital in Dublin. Ban on Catholics inheriting Protestant land
• Prohibition on Catholics owning a horse valued at over £5 (to keep horses suitable for military activity out of the majority's hands) Roman Catholic lay priests had to register to preach under the Registration Act 1704, but seminary priests and Bishops were not able to do so until 1778
• When allowed, new Catholic churches were to be built from wood, not stone, and away from main roads.
• Any and all rewards not paid by the crown for alerting authorities of offenses to be levied upon the Catholic populace within parish and county.

.

Because of the British policies which created these circumstances when the potato blight hit globally, the accompanying famine only hit the poorest, most vulnerable, who had been left systemically unprotected, the Irish.

Great Irish Famine
(Potato Blight) which ravaged potato crops throughout Europe during the 1840s. However, the impact in Ireland was disproportionate, as one third of the population was dependent on the potato for a range of ethnic, religious, political, social, and economic reasons, such as land acquisition, absentee landlords, and the Corn Laws, which all contributed to the disaster to varying degrees and remain the subject of intense historical debate.

.

Ireland too had other crops from larger farms which weren't in subsistence mode, but here too British policy dissuaded these crops from being used to ease the famine.

British had established import tariffs on grains which extended to Ireland, (Corn Laws). This made grains not subject to the potato blight costly to import, and it made Irish grains not subject to the British tariffs lucrative to resell in Britain.

Corn Laws (1815 - 1846) The Corn Laws were tariffs and other trade restrictions on imported food and grain ("corn") enforced in Great Britain between 1815 and 1846. They were designed to keep grain prices high to favour domestic producers, and represented British mercantilism, since they were the only mercantilist laws of the country.1 The Corn Laws imposed steep import duties, making it too expensive to import grain from abroad, even when food supplies were short.

While the population of Ireland was starving, Ireland exported much of there food through out the famine.

Irish food exports during Famine
Records show that Irish lands exported food even during the worst years of the Famine. When Ireland had experienced a famine in 1782–83, ports were closed to keep Irish-grown food in Ireland to feed the Irish. Local food prices promptly dropped. Merchants lobbied against the export ban, but government in the 1780s overrode their protests.[79] No such export ban happened in the 1840s.[80]

Throughout the entire period of the Famine, Ireland was exporting enormous quantities of food. In the magazine History Ireland (1997, issue 5, pp. 32–36), Christine Kinealy, a Great Hunger scholar, lecturer, and Drew University professor, relates her findings: Almost 4,000 vessels carried food from Ireland to the ports of Bristol, Glasgow, Liverpool, and London during 1847, when 400,000 Irish men, women, and children died of starvation and related diseases.

There were also cases where the British government interceded to discourage foreign governments from giving aid to the Irish during the famine.

Famine
Ottoman Sultan, Khaleefah Abdul-Majid I, declared his intention to send £10,000 to aid Ireland's farmers. However, Queen Victoria intervened and requested that the Sultan send only £1,000 because she had sent only £2,000 herself.

So the Sultan sent only the £1,000, but he also secretly sent five ships full of food. The English courts attempted to block the ships, but the food arrived in Drogheda harbor and was left there by Ottoman sailors. That £10,000 that the Sultan pledged to the Irish would be worth approximately £800,000 ($1.7m) today.

Historians Don't Agree

Wikipedia Irish Famine as Genocide

In 1996, Francis A. Boyle, a law professor at the University of Illinois at Urbana–Champaign, wrote a report commissioned by the New York-based Irish Famine/Genocide Committee, which concluded that the British government deliberately pursued a race- and ethnicity-based policy aimed at destroying the group commonly known as the Irish people and that the policy of mass starvation amounted to genocide per the Hague Convention of 1948.

.

In 1996, the U.S. state of New Jersey included the famine in the "Holocaust and Genocide Curriculum" for its secondary schools.

.

James S. Donnelly, Jr., currently retired emeritus professor of history at the University of Wisconsin–Madison, wrote in his book, "Landlord and Tenant in Nineteenth-century Ireland": .
I would draw the following broad conclusion: at a fairly early stage of the Great Famine the government's abject failure to stop or even slow down the clearances (evictions) contributed in a major way to enshrining the idea of English state-sponsored genocide in Irish popular mind. Or perhaps one should say in the Irish mind, for this was a notion that appealed to many educated and discriminating men and women, and not only to the revolutionary minority ... And it is also my contention that while genocide was not in fact committed, what happened during and as a result of the clearances had the look of genocide to a great many Irish. .
Cormac Ó Gráda (professor of Economic History and professor emeritus of economics at University College Dublin.) disagreed that the famine was genocide. He argues that "genocide includes murderous intent, and it must be said that not even the most bigoted and racist commentators of the day sought the extermination of the Irish", and also that most people in Whitehall "hoped for better times for Ireland". Additionally, he states that the claim of genocide overlooks "the enormous challenge facing relief agencies, both central and local, public and private". Ó Gráda thinks that a case of neglect is easier to sustain than that of genocide