The reputation of Aaron Burr has always suffered, mostly from his duel with Hamilton, and from the "Burr conspiracy" — Burr was put on trial for treason & acquitted, after which he exiled himself to Europe for a few years. He was in debt, struggled to find a new wife, and for a long time, his historical reputation was an extension of his disgraceful exit from politics. He has been framed as a womanizer, and sometimes deranged. And since Burr was an orphan, and he outlived his wife and daughter, and many of his papers were destroyed, nobody at the time or in later years had much interest in defending him.

Nancy Isenberg's Fallen Founder (2007) is notable because it examines Aaron Burr as a reaction to the "Founders Chic" biographies of the 1990s. The book seeks to dismantle those negative characterizations and give Burr the benefit of the doubt. It's fairly well-researched and very readable, although it can be dry at times (the first half before he enters national politics isn't terribly entertaining), & she is unable to reach a satisfying conclusion with the Burr conspiracy.

She does examine his relationship with Theodosia and his daughter, and notes that Burr was an early feminist who delighted in their intellectual pursuits, and a genuine politician with republican ideals. I would recommend the book for a more honest evaluation of Burr's character. You can also read her thoughts on Hamilton the Musical here, and similar articles that are critical of Miranda's portrayals are widely available.

But if anyone in Hamilton is sneaky, it's Hamilton himself. While Jefferson is also guilty of gossip, Hamilton went even further, spreading rumors and working behind the scenes, especially as leader of the Federalist Party. In the Election of 1796, he also floated Vice President Thomas Pinckney as an alternative to John Adams who could win more votes in the South. Historians looking backwards from 1800 have even alleged Hamilton was conspiring against him in 1796, but this is uncertain.

John Adams eventually had to replace his original cabinet members, who were loyal to Hamilton and would write letters to him in New York. The party split between Hamilton's more conservative faction, the High Federalists, and Adams's moderate faction. In 1800, Hamilton hurt Adams's re-election chances when his 54-page criticism of Adams fell into the wrong hands and was circulated by the Jeffersonians.

None of this is addressed in Hamilton, and in fact Adams doesn't even make an appearance, because it would be quite difficult to portray Hamilton as the main character of that story or cast him in a sympathetic light. Instead it shifts over to the Reynolds Pamphlet, since you can at least feel bad for the guy being blackmailed.

Hamilton was a very skilled Secretary of the Treasury, but he was also a militaristic elitist with a rather explicit distaste for popular democratic government. Instead, he specifically wanted the power of the strong national government to reside in the upper class & business interests, which would lend it stability and protect it from the lower classes. In describing the two classes of people, he writes:

The first are the rich and well born, the other the mass of the people. The voice of the people has been said to be the voice of God; and however generally this maxim has been quoted and believed, it is not true in fact. The people are turbulent and changing; they seldom judge or determine right. Give therefore to the first class a distinct, permanent share in the government. They will check the unsteadiness of the second, and as they cannot receive any advantage by a change, they therefore will ever maintain good government. Can a democratic assembly, who annually revolve in the mass of the people, be supposed steadily to pursue the public good? Nothing but a permanent body can check the imprudence of democracy. Their turbulent and uncontrolling disposition requires checks. (Hamilton, 1787)

This is hardly addressed in Hamilton, and in my view, the musical could have been much better if it wasn't trying to be a story of the Revolution and a story of the Federalist Era and a revival of Hamilton as a figure at the same time. I would've loved to see just pure mudslinging, no holds barred. And in my opinion, the musical fails to address the honestly very entertaining and harsh political climate of the Federalist Era, doesn't really set up the stakes or the characters it's presenting you, and while it pays good attention to its female characters, the way it resolves slavery is offputting. (That is to say, it gives slaveowners black representation in the form of Thomas Jefferson, gives characters brownie points for their short-lived antislavery measures, and calls it a day.)

To end with some proper mudslinging, Adams wrote this of Hamilton in 1806, in a letter to Benjamin Rush about "the Vapours."

"What a pity it is that our Congress had not known this discovery, and that Alexander Hamiltons project of raising an Army of fifty thousand Men, ten thousand of them to be Cavalry and his projects of Sedition Laws and Alien Laws and of new Taxes to Support his army, all arose from a superabundance of secretions which he could not find Whores enough to draw off? and that the Same Vapours produced his Lyes and Slanders by which he totally destroyed his party forever and finally lost his Life in the field of honor."