I had wrote a Hamilton-wank post a while back where he gets the Elected Monarchy he wanted and basically becomes a king. Hamilton would probably be canny enough to keep things together during his lifetime, but the problem with concentrating a lot of power in one man is the vacuum that opens up after they die, and the second act that follows them. Sometimes you get someone else just as competent and it works out (eg Caesar/Augustus), but a lot of times it doesn't (eg Bismarck/Wilhelm II), or it just falls apart (eg post-Napoleonic France).
Say Hamilton dies around 1820 after a decade-plus of ruling as a President-king. His apparatus of government is all going to be packed with Federalists, and the Democrat base shrivels up after a decade or so of being pretty much non-competitive. All the original Founding Fathers are either gone or nearing the end of their line, so without that big-name opposition, the budding Federalist machine "nominates" a bunch of younger Federalist toadies to run for king (a funny phrase for sure..). This new King openly plays the patronage and politics game, the army starts cozying up to the only Commander in Chief they'll have for the next 30 years, and the Republic veers towards a one-party dictatorship.
Having one party permanently in power doesn't always mean you're going to get some kind of dystopian fascist nightmare - Mexico held it together OK for 80 years - but it does open the door to get pretty abusive. If you want a Protestant state church to disenfranchise the Catholic Irish that will start coming in the 19th century, you could do that. If you want to shut down your political enemies in the press, you could do that. Democrats claiming violation of their 1st Amendment rights won't get a very sympathetic ear in a court system packed with nothing but Federalist-appointed judges.