Before his son Philip died in in 1801, Hamilton was, for all the years he did his work "Founding" America, like the other key Founders, a theistic rationalist.
Douglas Adair and Marvin Harvey wrote an excellent article in the William & Mary Quarterly in 1955 entitled Was Alexander Hamilton a Christian Statesman? Adair and Harvey identify four religious phases in Hamilton's life: He had a conventionally religious youth. From 1777 to 1792, he seemed totally indifferent to religion. From the period of the French Revolution onward, he had an "opportunistic religiosity", seeking to use Christianity for political ends, and then after the death of his son Philip in 1801, truly became a repentant orthodox Christian. The first three phases of his life were consistent with theistic rationalism.
In the Farmer Refuted he spoke the following like a true theistic rationalist:
The sacred rights of mankind are not to be rummaged for, among old parchments, or musty records. They are written, as with a sun beam, in the whole volume of human nature, by the hand of the divinity itself; and can never be erased or obscured by mortal power.
During the period from 1777 to 1792 is supposedly when he made two wisecracks about God. When asked why God wasn't mentioned in the US Constitution Hamilton supposedly said, "We forgot." The other was Hamilton supposedly didn't agree to Franklin's call for prayer because he didn't think the Constitutional Convention needed "foreign aid." Both of these may turn out be apocryphal. They are, as David Barton would put it, "unconfirmed" in the primary source record.
However, during that period, he said something arguably much worse. From 1777 to 1792 there are, according to Adair and Harvey, only two letters where Hamilton mentions God or religion at all. One of them, a letter to Anthony Wayne July 6, 1780, he discusses a military chaplain:
“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."
[Here is Ron Chernow discussing the quotation in his award winning book on Hamilton.]
I've been challenged with a version of that letter that didn't include the "whore" part. Adair and Harvey inform us exactly of the whereabouts of the original and that the Henry Cabot Lodge version from 1904 had been "bowdlerized" probably to make Hamilton look not so creepy. I've uploaded the page from Adair's and Harvey's article with the information about the quotation in question:
Is this the kind of thing a "Christian Statesman" says? Hamilton had a reputation for being a rake among other Founders as well. John Adams called Alexander Hamilton a “bastard brat of a Scotch pedlar,” who had a “superabundance of secretions which he could not find whores enough to draw off.” Finally Adams decried “the profligacy of his life; his fornications, adulteries and his incests.”
Abigail wasn’t much nicer. “Oh, I have read his heart in his wicked eyes. The very devil is in them. They are lasciviousness itself.”
And of course Hamilton was caught having an affair with Maria Reynolds, a grifter. And he made a self-serving, less than forthcoming apology about the matter.
Hamilton's only other reference to religion in the period from 1777-92 concerned what he desired in a wife. As he wrote to John Laurens in Dec. 1779: "As to religion, a moderate streak will satisfy me. She must believe in god and hate a saint." Certainly not the words of a devout orthodox Christian. (Ironically, the woman he ended up marrying, Eliza Schuyler, was devoutly religious.)
As noted, after his son died in 1801 Hamilton did begin to display, for the first time in his public career true humility and a likely then became orthodox Trinitarian Christian. I say there is evidence that he was a Christian because instead of talking about "the Deity" in generic or philosophical sense, for the first time in his public life he uses actual specific Christian language about God. Though, he was refused communion at his death, which he begged for, because he hadn't yet joined a Church -- more evidence of his being a "newbie" Christian at the end of his life.