Why is King Henry V regarded as such a great King?

Arguably, Henry V laid the foundation for a united, strong, modern England.

After a misspent youth, he put down rebellions against the English crown by Percy, the "Hotspur" of the North, and Glendower of Wales, another dissaffected area. This was basically the last time that "England" threatened to fall apart.

Overseas, his victories at Agincourt and elsewhere almost allowed the English to claim the French throne. In the minds of many historians, these achievements outweighed the brutality by which they were accomplished (which was practiced by others). Subsequent events (e.g. Joan of Arc) prevented the conquest of France, but Henry V gets credit for "trying."

Henry also launched expeditions further afield to help the Teutonic Knights defeat the Lithuanians. again, they were not successful in their original purposes, but did put England on the map, because they demonstrated the degree to which England was able to project power.

Great is not the same as good.

If you look at almost all great men, especially ones who won their primary glory on the field of battle, there are few if any who do not have dark deeds attached to them.

Partly this is because brutality and atrocities were more common and more accepted historically than they are now - the conventional fate of besieged cities for centuries is a prime example. [If a city under siege refused to surrender after a breach was made in the walls, it would be sacked in the following assault. This was pour encourager les autres, so to speak]. Partly, it is because those who won great reputations for themselves were often more ruthless than their fellows - the first example of this in my mind is Cromwell. An example of Henry's ruthlessness would be the murder of prisoners at Agincourt which you mention; as most sources seem to agree that he ordered it when he was concerned that the (thousands) of prisoners might overwhelm their guards, take up weapons and pose a severe threat to his army. Their murder remains unacceptable; however, his ruthless action ensured the completeness of his victory on that day.

And of course this is not to forget the final aspect - that many people are uncomfortable with flawed heroes, so once an historical figure is being held up as generally admirable, their mis-steps and wrongful actions can often be de-emphasized and brushed under the carpet.

In Desmond Seward's book on the hundred years war, he introduces Henry thus:

In the national legend Henry V remains the most heroic of English Kings. He is the glorious conquerer who broke the French chilvary at Agincourt and won the throne of France for his son's inheritance.

Henry V is obviously best known for his military conquests. His military career began in his teens, fighting successfully against the Welsh rebel Owain Glyndwr and his allies the Percies and Mortimers. As king, he won a famous victory at Agincourt and in later campaigns, so successfully conquered territory that he forced the French into the Treaty of Troyes that stipulated that the French crown should pass to his line upon Charles VI's death.

Henry's personal and political skills also did much to cement his reputation. He had a much better relationship with parliament than did his profligate father and other predecessors. He was able to put aside personal differences, willing to appoint to powerful positions the sons of men who had been executed by his father and winning their loyalty. Indeed, it was Edwin Mortimer, former heir of the deposed Richard II, who informed Henry of the Southampton plot to depose Henry and replace him with Edwin. Also, Juliet Barker wrote that he was particularly scrupulous in ensuring that he did everything possible to ensure that there was divine justification in his military actions (but her Agincourt book does tend to read as a panegyric to Henry).

Henry predeceased Charles, leaving his son Henry VI to inherit the French crown and his brother John as regent to take England to the peak of its power in France before its inevitable fall. Perhaps Henry V's early death helped his reputation, as he lacked the opportunity to suffer the reverses that had afflicted Edward III and the Black Prince after the successes of Crecy and Poitiers early in their careers.

Culture undoubtably played a role in Henry's reputation. As with Richard III, the public's impression of Henry V owes much to the works of Shakespeare, so we are thus left an image of Henry both as heroic conquerer and (unfairly) as dissolute youth. Shakespeare, as usual, borrows from earlier works. Barker wrote that:

As the last vestiges of English power in France were slowly but inexorably eradicated, people looked back to the glory days of Agincourt with nostalgia. Ballads, chronicles and plays in English were written for an increasingly literate bourgeoisie preserved the memory of the victory and served as a rallying cry for future wars in France.

The need for a pristine image of Henry V to serve as an English propaganda tool (including as recently as WW2) has led to some of his more questionable acts being glossed over. I'll leave the last word to Seward's final sentence of the introductory paragraph quoted above:

In reality he displayed a number of markedly unheroic qualities and, in a gentlemanly, medieval sort of way, he had more than a little in common with Napoleon and even Hitler.