Hamilton was ineligible for election to the office of the Presidency.
Actually he was. You need to either be a natural born Citizen, or a Citizen of the U.S. as of the adoption of the Constitution to be president. While Hamilton does not meet the first possibility, he does the second as he was citizen of the United States as of 4 March 1789.
What I was thinking about, primarily, when I made this, was Hamilton's foreign policy given the trouble with the French. I'm wondering if this establishes an early precedent for American foreign intervention, and maybe even makes Conservatives warhawks earlier on than in OTL. Also, how America entering into war so early might effect it. How might these questions be changed with a Hamilton Presidency beginning in 1805?
Not nearly as much as they would have if he'd become President in 1797. By 1805 it was clear that France was not going to be a bastion of republican virtue. If Hamilton had tried to side with Britain in 1797, it could well have led to the breakup of the United States. In 1805, war with France would have been a unifying factor both north and south. The south would have had Louisiana and Cuba to gain from the war, while the north would have been able to maintain the more profitable trade with Britain.
Conversely, uninterrupted trade with the British will retard the development of Yankee manufactures.