Based on the letters and writings we have from Hamilton and Eliza Schuyler, was there any evidence of the founding father having any ambitions to become a President after George Washington?
The Presidential eligibility clause:
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; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
We cannot say definitively one way or the other, but evidence suggests maybe. Hamilton wrote a draft Constitution that included language making Hamilton eligible:
No person shall be eligible to the office of President of the United States unless he be now a Citizen of one of the States, or hereafter be born a Citizen of the United States.
In that same article from Columbia Law the author writes:
Had Hamilton had genuine plans to run for the Presidency in the future, it seems extremely unlikely that he would have suggested standards for presidential eligibility which would have excluded him from consideration, standards which were eventually fully adopted in substance,
This makes sense.
However there is no evidence that Hamilton included "unless he be now a Citizen of one of the States" specifically for himself or for all immigrants in the same situation.
John T. Morse Jr. in his 1876 The Life of Alexander Hamilton wrote - page 223 -:
But Hamilton appears never to have aspired to this office [the Presidency], never to have been brought by his friends into this arena.
So at best, inconclusive.
It may incompletely answer your question, but /u/Chronic_Apathy1 gave this answer to a similar question during the Hamilton AMA on this sub.