I am planning on keeping Washington alive, although not past 1805. He was pretty old for the time when he died OTL (67 years?).I doubt that Hamilton would consider breaking Washington's 2 term policy. Hamilton realized that he could nor ever should break a precedent set by Washington. We would likely see a 2 term Hamilton that remains very active in Federalist Policies, even possible become Governor or Senator from NY. Plus with Hamilton as President i could see Hamilton making Washington Head General of American forces causing Washington to live longer because he doesn't catch pneumonia.
You need to figure out who will be in Hamilton's Cabinet
As for Hamilton's cabinet, I could see J.Q.A. possibly becoming Secretary of State, but I need to look into his diplomatic career pre-1800.