What was the original process for becoming a US citizen?

tl;dr: Common Law, inherited from Britain, says you're a citizen by right of birth or parentage... but a citizen of what? The principles of the US revolution imply your first obligation is to your society (ie. the people of your state). When your state changes its allegiance, so do you. An analogy can be drawn to if your state rewrites its constitution: the rules change, but you're still a citizen of that state.

While the US Constitution makes mention of "a Citizen of the United States", it does not define what that means. For example, in Article II, Section 1 it outlines the requirements for President including...

No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President

And again in Article II, Section 2 with the requirements for being a Representative.

No Person shall be a Representative who shall not have attained to the Age of twenty five Years, and been seven Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State in which he shall be chosen.

This became a subject of debate shortly afterward when David Ramsay petitioned James Madison over whether William Smith of South Carolina could be a representative. Despite having been born in Carolina, David claimed he had not been a citizen for seven years because he left the US from 1770 to 1783.

As in the time of his absence the Revolution took place I contend that in order to his becoming a Citizen of the United States something must have been done previously on his part to shew his acquiescence in the new Government established without his consent. The lowest test of acquiescence is in my opinion residence in the Country. Till he resided under the Government of the united States I cannot therefore see how he acquired Citizenship. We were all born subjects [of the King of Britain] but you and I were released from our allegiance by the restraining act of Parliament passed in December 1775. You and I became Citizens by being parties to the Declaration of Independence. By that act a new compact for a new Government was formed between the then residing and consenting inhabitants of these States. But an absent native neither lost his allegiance by the one nor acquired Citizenship by the other. Such continued subjects while in Europe and under British protection and could only become Citizens on their returning and by residence by an oath or by some other mode manifesting their acquiescence in the revolution.

Basically, Mr. Ramsay is arguing that for someone to be a US citizen it requires both release from their obligations as a British citizen and the active consent of the person to become a citizen of the US; that citizenship is a explicitly consensual act between the person and their chosen sovereign. He argues residents in the US during the war are assumed to have consented to all this because they stayed in the US, but you can't make that same assumption about someone living in Europe. He's arguing Mr. Smith was still a British subject at least until he returned to the US, and should probably take an oath or something to be extra sure.

James Madison took the petition before the House. His reply to Mr. Ramsay outlines the Congressional thinking about citizenship at the time. Madison starts by admitting that this issue is legally hazy.

It were to be wished, that we had some law adduced more precisely defining the qualities of a citizen or an alien; particular laws of this kind, have obtained in some of the states; if such a law existed in South-Carolina, it might have prevented this question from ever coming before us; but since this has not been the case, let us settle some general principles before we proceed to the presumptive proof arising from public measures under the law, which tend to give support to the inference drawn from such principles.

You can see here that the people still consider themselves citizens of States first, and then those states are part of a Union. This is implied in several places in the Constitution, Article IV, Section 2 governing interstate law is one such example.

The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.

A Person charged in any State with Treason, Felony, or other Crime, who shall flee from Justice, and be found in another State, shall on Demand of the executive Authority of the State from which he fled, be delivered up, to be removed to the State having Jurisdiction of the Crime.

Mr. Madison then goes on to quote Common Law inherited from Britain that Citizenship is granted by the location of your birth, or sometimes by your parents. This goes back to serfdom, peasants were tied to the land. Now it's being asserted with some pride.

It is an established maxim that birth is a criterion of allegiance. Birth however derives its force sometimes from place and sometimes from parentage, but in general place is the most certain criterion; it is what applies in the United States; it will therefore be unnecessary to investigate any other. Mr. Smith founds his claim upon his birthright; his ancestors were among the first settlers of that colony.

Then Mr. Madison goes on to assert the revolutionary idea that your allegiance to the sovereign and to the people (he uses the term "society") are different, in and of itself radical at the time, but goes further to say that your allegiance to the people is the higher one. Not unexpected from a people who just rebelled against their sovereign in the name of the people.

[David Ramsay] supposes, when this country separated from Great Britain, the tie of allegiance subsisted between the inhabitants of America and the king of that nation, unless by some adventitious circumstance the allegiance was transferred to one of the United States. I think there is a distinction which will invalidate his doctrine in this particular, a distinction between that primary allegiance which we owe to that particular society of which we are members, and the secondary allegiance we owe to the sovereign established by that society.

Madison goes on to explain that the King could not make you a citizen. You were a citizen either by birth or by act of Parliament, and that this implied allegiance to the nation first.

This distinction will be illustrated by the doctrine established by the laws of Great Britain, which were the laws of this country before the revolution. The sovereign cannot make a citizen by any act of his own; he can confer denizenship, but this does not make a man either a citizen or subject. In order to make a citizen or subject, it is established, that allegiance shall first be due to the whole nation; it is necessary that a national act should pass to admit an individual member. In order to become a member of the British empire, where birth has not endowed the person with that privilege, he must be naturalized by an act of parliament.

Mr. Madison then asserts that because a citizen foremost owes their allegiance to their society, in this case the state of South Carolina, their citizenship transferred when South Carolina transferred its allegiance from Britain to the US. William Smith was indirectly a subject of the King through his citizenship of South Carolina.

When that society [South Carolina] separated from Great Britain, he [William Smith] was bound by that act and his allegiance transferred to that society, or the sovereign which that society should set up, because it was through his membership of the society of South-Carolina, that he owed allegiance to Great Britain.

Finally, Mr. Madison states that while the bonds of allegiance between the States and the King was dissolved, the bonds between the States and their Citizens were not.

This reasoning will hold good, unless it is supposed that the separation which took place between these states and Great Britain, not only dissolved the union between those countries, but dissolved the union among the citizens themselves: that the original compact, which made them altogether one society, being dissolved, they could not fall into pieces, each part making an independent society, but must individually revert into a state of nature; but I do not conceive that this was of necessity to be the case; I believe such a revolution did not absolutely take place.

He backs up this argument with an analogy: if South Carolina rewrote its state constitution you'd still be a citizen of South Carolina, just under new rules.

Suppose the state of South Carolina should think proper to revise her constitution, abolish that which now exists, and establish another form of government: Surely this would not dissolve the social compact. It would not throw them back into a state of nature. It would not dissolve the union between the individual members of that society. It would leave them in perfect society, changing only the mode of action, which they are always at liberty to arrange.

Mr. Madison goes on to assert that because William Smith was a minor at the time of the revolution, he wasn't fit to consent. Rather he is compelled to be bound by the decisions his society has made in his stead.

If he was not a minor, he became bound by his own act as a member of the society who separated with him from a submission to a foreign country. If he was a minor, his consent was involved in the decision of that society to which he belonged by the ties of nature...

Mr. Smith being then, at the declaration of independence, a minor, but being a member of that particular society, he became, in my opinion, bound by the decision of the society with respect to the question of independence and change of government;

Eventually the rules of US Citizenship would be defined by law, the first of which being the Naturalization Act of 1790 defining how to become a naturalized citizen. This was allowed by Article 1, Section 8 of the Constitution, but Congress had to define the rules.