I can see a couple general ways it could happen.
First, a sustained breakdown of government under the Constitution would mean all bets are off. At that point, it becomes relatively easy for whoever restores order to become a
de facto monarch, as happened with Oliver Cromwell and Napoleon Bonaparte. Possible circumstances for this to happen:
- The Continental Army seizes power from the Continental Congress shortly after the Revolution, as some of Washington's officers considered urging him to do IOTL. This would require either a major change to Washington's character or a different commander of the Continental Army.
- Hamilton attempts a coup against Adams in 1798-99, possibly triggering a multi-sided civil war. No evidence AFAIK that Hamilton seriously considered doing so, but Adams seriously feared that he would.
- A major war with a European power that goes badly enough to be an existential threat to the United States, especially if it involves foreign occupation of a large section of the US, invasion preventing either elections or congressional sessions, or attempts by US states to make a separate peace with the enemy. Perhaps a War of 1812 that goes considerably worse for the US than OTL or a Trent War scenario.
Alternatively, you could have a monarchy evolve organically from Constitutional structures. The President's powers are modeled pretty closely on the powers of the King of Great Britain at the time; the major differences are the mode of selection (election for a term of four years rather than inheritance for life), moving the power to declare war to the legislature, and giving the legislature the ability to override a veto. If people somehow get in the habit of electing the same family over and over again, as with House of Orange's role in the First Dutch Republic, you get a
de facto Monarchy and eventually it becomes easy to drop the pretense and create a formal monarchy.
The major obstacles to this are Washington's two-term tradition and the effects of the two-party system. The former for obvious reasons, and the latter because the existence of a rival party makes it hard for a consensus to emerge in favor of near-automatic reelection. Possible scenarios:
- The two-party system never emerges. Get both Hamilton and Jefferson out of politics early in the Washington administration, and the Federalists may be able to hold together as a near-consensus party (Madison and many moderate Democratic-Republicans started out as Federalists, and only left out of opposition to Hamilton). If Washington then dies in office, there's no two-term tradition, and President Adams would have a good shot at repeated reelection as well as a son who's Presidential material in his own right.
- An extended Era of Good Feeling. Perhaps Monroe runs for a third term and goes unopposed as he did when he ran for his second term, and then goes on to win a fourth term as well before dying in office. He'd need to have replaced Tompkins as VP with a more plausible successor (probably William Crawford; JQA or Clay are plausible presidents, but would likely be too controversial to avoid the emergence of an opposition party) who could keep the Era of Good Feeling going until people come to take it for granted. Keep it up long enough, and eventually you'll get a President setting up his son as VP and successor.
There are opportunities for later breakdowns in the two-party system and the two-term tradition (the best being Grant or Teddy Roosevelt), but by then I think it's too late to fundamentally change American political culture to the extent necessary for a monarchy to emerge organically; more likely, the two-term tradition and the two-party system would reassert themselves as they did after they partially broke down under FDR.