In the musical (which is fantastic for those who haven't listened to it), their relationship is depicted in a very dramatic arc, so much that I can't really believe it reflects reality. Here are the basic strokes of he musical for those who don't know:
Hamilton looks up to Burr starting in the early 1770s, who always advises him to never show his opinions. Hamilton pushes back and always is very forthright and thereby becomes a leader in the revolution while Burr is more circumspect. They remain friends despite this, even though the musical makes it seem like everyone dislikes Burr, except for Hamilton. They practice law together after the revolution.
Burr is shown as being calculating and without any true beliefs, doing anything solely for political gain and with no regards for principle. He refuses to help write he Federalist Papers. He then defeats Hamiltons father in law for a Senate seat, harming their relationship.
Their relationship degrades when Burr uses the Reynolds affair to threaten Hamilton, threatening exposure of the affair if Hamilton crosses him.
Finally, Hamilton endorses Jefferson in 1800 over Burr, causing Burr to challenge Hamilton to a duel.
So how accurate is all this? What big picture info is missing that is really important for someone interested in the story to understand?
It's pretty unlikely that Burr and Hamilton were friends as young men. Burr, a descendant of Jonathan Edwards and as close to an aristocrat as early Americans came, went to the College of New Jersey (later Princeton). Hamilton was an illegitimate child born on the West Indies island of Nevis. He was eventually sent to America by the man rumored to be is biological father where he attended King's College in New York (later Columbia).
It's likely that they met during the war, specifically during the British campaign to capture New York. They both served under General Putnam in the rear-guard left to defend the city itself while the main army retreated. They only narrowly avoided capture themselves, and in fact witnesses later recalled that Burr led the artillery men (including Hamilton) away from the British just in the nick of time.
I'm drawing from Michael Newton's book on Alexander Hamilton and Robert Middlekauff's book on the Revolution call The Glorious Cause.
Thanks for the question about Hamilton! I've been obsessed with this musical and would love to see it discussed by our historians. If I may contribute a follow-up-ish question, I am curious to hear opinions on Ron Chernow's biography which the musical is based on, if anyone has happened to read it. I am assuming that the facts in the book are pretty close to reality, esp since Lin-Manuel was very concerned with being as historically accurate as possible, but i don't know. Is this question allowed?
I also love the musical, and recently finished Ron Chernow's biography which the musical is based on.
While it's possible Aaron Burr and Alexander Hamilton had run ins during the war and the immediate aftermath, they weren't notable acquaintances.
Burr was in Hamilton's sphere due to his intervention and diplomacy when Hamilton and Monroe came close to dueling over the Reynolds affair. Unlike the musical, the reveal over the affair likely came from Monroe, who was privy to Hamilton's confession (not Burr, Thomas Jefferson, and James Madison). Monroe agreed to destroy evidence and instead passed it along. Also interesting, Burr represented Maria Reynolds in her divorce from James Reynolds.
Burr was also involved in a duel with John Baker Church, who married Hamilton's sister-in-law Angelica. He was cavalier during this duel and not intent on killing Church, which possibly lead to Hamilton under-estimating his intent in their duel.
While the musical takes some liberty with their relationship, the spirit of it is largely on point.
In the musical they have Burr and Hamilton as seconds at the duel between Charles Lee and John Laurens.
Which is excellent foreshadowing.
But in reality Charles Lee's second was not Burr, it was Evan Edwards.
Burr didn't challenge Hamilton after the election of 1800, but after Burr's attempt to secure the governors mansion in New York in 1804. Hamilton did make significant opposition to his election and Burr's challenge could have been related to that, but the issue Burr raised was due to a letter from a Charles Cooper alledging that Hamilton had made statements disparaging to his character and alluded to "a more despicable opinion" expressed but not specifically stated. Burr asked Hamilton to refute this more despicable statement, but Hamilton could not given that he wasn't told what statement he apparently made. After several exchanges the bulk of the communication is handled by Burr and Hamilton's seconds, Van Ness and Pendleton, respectively. You can read the full text of the exchanged letters leading up to the duel here.
https://archive.org/stream/burrhamiltonduel00gayl/burrhamiltonduel00gayl_djvu.txt
Fair warning it is in fairly obscure 1800s lawyer speak but the subtext underneath the flowery language definitely has an acrimonious tone to it.
Whether they were friends or not? I think that it might be closer to respected adversaries on the political arena and they each had respect for each others military achievements.