I agree with
@GlobalHumanism insofar that I don't see such a thing happening in OTL's United States, because the USA was far too stable for it. Even if the Constitutional Convention had failed utterly, the Articles of Confederation would just have been amended, instead. Things like Shays' Rebellion were dealt with in OTL. So I don't see the basis for a 'Napoleonic' coup. But let's consider a scenario where the situation is less stable from the outset:
In a thread about disintegration of the early USA, I expressed the idea that this could best be accomplished by installing some kind of dictatorship after a military coup-- which would then be overtrown, leading to most states fearing centralised power, and essentially falling apart into separate countries (or multiple regional unions). My scenario was that George Washington is killed relatively the war, someone else - with less skill - assumes command of the military, and the war drags on longer. Congress, in OTL notorious for its failure to act in a timely manner (and often failing to adequately supply and pay the army), would be even more vehemently criticised in this ATL. Inflation was already a major problem in OTL, and would be worse in case of a longer war. Eventually, either the war is won but the country is utterly bankrupt (so veterans get no backpay and pensions, and Congress is widely criticised) or the war drags on so long that Congress is simply no longer seen as fit to govern by the military. In either scenario, one or more military leaders seize power to "rectify the situation".
That, of course, was about the specific goal of getting the Union to fall apart at the very outset. The idea was that the coup would be a disaster, the 'tyrants' would be widely hated, leading to the defeat of this centralising, authoritarian military dictatorship (and thereafter, the disintegration of the USA). But... the military coup doesn't
have to lead to a terribly oppressive tyranny, of course. One might well argue that in this scenario, Congress
is failing. So suppose instead that one or more generals seize power, with the explicit aim of creating a more stable (and thus, more powerful) central government. But that they are not very oppressive, and are - like Napoleon in OTL - mostly seen as a bit of a relief for a troubled country.
As for the nature of the regime: I get the impression that most of the generals and top officers in OTL tended towards the (proto-)Federalist kind of politics, and that fits with the aim of strong government, economic controls to fight a chaotic situation, etc. -- So while the exact nature of the resulting government would depend very much on who, exactly, seizes power, I'd expect a man like Alexander Hamilton to be very happy about the results (and quite possibly to - later on - have a hand in shaping them).