As to his creed, no evidence shows he was a "mere Christian" (as CS Lewis would term it, believing in a Triune God, etc.) until after he had done his work Founding America, after his son Philip died in a duel, after his life came crashing down on him.
Hamilton was such a newbie orthodox Christian when he died, he never got around to joining a church and was initially refused communion by both of the orthodox Churches with which he sought communion.
Hamilton always seemed to have been a theist, and like Washington and most other Founders believed in the value of "religion" (which included orthodox Christianity) for its instrumental purposes, the way it facilitated republican virtue. During the later 1790s he did seem to lament the way cold, strict deism headed towards atheism during the French Revolution.
He nonetheless does NOT claim, during those periods, that orthodox Trinitarian Christianity is the only way to God or that this is the only form of "religion" that properly supports republicanism. Hamilton's opinions on the French Revolution and its turn towards irreligiousity seem exactly those of and completely compatible with the militant unitarianian, John Adams'. Note, I do not say this proves Hamilton was a "unitarian" like Adams during this period, just that it does not prove, as some argue (i.e., the Christian America narrative), Hamilton was orthodox during this period or he defended only "real" orthodox biblical Christianity because he was one of "them."
In order to believe in the "Christian America" narrative of Hamilton's orthodox Christianity during the 1790s one has to read in one's desires that do not exist in the historical record.
During the Constitutional Convention, it was rumored, when asked why they didn't mention God in the Constitution, Hamilton said, "we forgot." And that they didn't need "foreign aid." Those two quotations appear to be "unconfirmed" as David Barton would put it.
I like Matthew Franck's reaction when learning those quotations were "unconfirmed":
... I doubt very much that Alexander Hamilton was ever such an ass as to say such a thing, even in his most freethinking days. Nothing like it appears in Hamilton’s collected papers. ...
I disagree that "nothing like it" appears in Hamilton's record. If that's your standard for what constitutes an "ass," other things Hamilton said during that period arguably merit him that label. For instance, when speaking on what he values in a military chaplain, in a letter to Anthony Wayne July 6, 1780, Hamilton said:
“He is just what I should like for a military parson except that he does not whore or drink. He will fight, and he will not insist upon your going to heaven whether you will or not."
Likewise when noting what he was looking for in a wife, to John Laurens, December, 1779, Hamilton writes:
In politics I am indifferent what side she may be of. I think I have arguments that will easily convert her to mine. As to religion a moderate stock will satisfy me. She must believe in God and hate a saint.
But as to fortune, the larger stock of that the better. You know my temper and circumstances and will therefore pay special attention in the treaty. Though I run no risk of going to Purgatory for my avarice, yet as money is an essential ingredient to happiness in this world, as I have not much of my own, and as I am very little calculated to get more either by my address or industry, it must needs be that my wife, if I get one, bring at least a sufficiency to administer to her own extravagancies.
Notice how Hamilton cares more about the specifics of his political creed than his religious creed. He cares about converting his future wife to his politics; but as to religion, she has to merely "believe in God and hate a saint." We see no, "I have arguments that will easily convert my wife to Christianity because I can show her the evidences of its historical truth."
Given his wife turned out to be an orthodox Christian, it's likely that she converted him at the end of his life.
Finally, regarding Hamilton's self interested avariciousness referenced in the above letter, if compatible with "Christianity" at all, seems more like the prosperity Gospel of Benny Hinn, Jim Bakker, or Robert Tilton than of respectable Christianity.
[Final final note: I'm not so sure how seriously to take Hamilton's crack on "purgatory." That's certainly part of Roman Catholicism, an orthodox faith; but most reformed/evangelical Protestant creeds of the Founding era, like those today, reject purgatory. The more "enlightened" Protestant Christian unitarian-universalists, however, did believe in Protestant Purgatory, where good people went to Heaven, bad people were temporarily punished there.]