Well, I've consumed the (generally excellently-reviewed) Ron Chernow biography that was Miranda's jumping off point, and he summarizes Chernow's findings pretty well, albeit with much condensing for the stage. Burr leapt between the two nascent parties for what seemed to be expediency's sake. He was cheerfully duplicitous - the most jaw dropping example to me was the Manhattan Company, which Burr proposed as a private water provider for New York City (which was then suffering from outbreaks of yellow fever) and which he... Ugh it's super complicated, but the gist is he got tentative approval, then changed the language of the final proposal that went to the governor for his signature, all without telling anyone. Surprise, his "water company" was essentially also a bank, chartered far more leniently than the actual banks of the day. Oh, and he removed the provision that the company would provide water to those in need during such crises as the one New York was experiencing. His partial motivation was to stick it to the Federalists, but his maneuver was seen as being beyond the pale to many of the Republicans he was ostensibly courting.

In short, if we take Chernow as a reliable source, Burr was indeed a fairly craven politician, a pawn in search of patronage. Now, it's a separate question as to whether he actually held any beliefs strongly. He was characterized, including by himself, as being incredibly reserved and discreet with his opinions. He seemed all too aware of the benefits of being immune to criticism if he remained a blank slate.

If the subject interests you, Chernow's book was a very good read IMO.

Edit to add:

Regarding what he campaigned on, actually he did heavy political lifting in New York elections that would determine representatives to the electoral college. When his efforts paid off in helping to flip the City's 12 electors from Federalists to Republicans, he had earned political capital that was then cashed in to put him on the Republican ticket, explicitly as vice president. He seemed content with that role. Due to the vagaries of his the system worked then, he and Jefferson ended up in a tie. While he initially made the proper noises that he would defer to Jefferson, he eventually changed course and made it known he would accept the presidency. In all of this, his standing fire any particular issue isn't really how it was done. Campaigning in general seems to have been disdained as ungentlemanly, and while Burr was ahead of the curve on that subject he still didn't go about it in ways we would generally recognize today.

Edit 2: I'm pretty excited that after years of lurking this sub I think i actually have something useful to contribute. :)

Edit 3: changed "Federalist ticket" to "Republican ticket".

So I'd like to piggy-back off of /u/Pitchwife 's answer above that clearly articulate's Burr's persona and characteristics post Revolutionary War according to Ron Chernow. While I think his biography is quite fair in his depiction of Burr, not all Revolutionary scholars will agree.
There are some indisputable facts (Yes, Burr did switch parties primarily so he could unseat Philip Schuyler from office for a seat in the New York State legislature).

Fallen Founder: The Life of Aaron Burr by Nancy Isenberg shares many of the same views around the facts of Burr's life, however she makes some key distinctions. While she concedes that Burr at times was willing to concede on many of his political stances, Isenberg believes that Burr had several genuine political principals that he would not waver away from. She zeros in on two in particular that she believes he would never waver on: women's education and the ability for "popular" participation in government.

This comes a bit in contrast to both Hamilton: The Musical (which I'm a huge fan of!) and Chernow's biography which both argue that Burr just wanted to be in the political spotlight for selfish ambitions rather than having any true political position worth pursuing.

Now to address your question specifically, many of Burr's beliefs aren't as straightforward as one might hope. Check out Isenberg's description of Burr's views on slavery:

Yet Burr's views on slavery were not simple. He represented two manumitted slaves before the New York Court Chancery in what must have been a controversial case... During his short term in the state assembly in the 1780sBurr went so far as to propose a radical revision to a bill to abolish slavery, calling for the immediate emancipation for all slaves. He opposed three different amendments that restricted the right of free blacks to vote, to serve on juries, or to testify against white -- and found a status that penalized a black 100 (pounds) for marrying a white. His position seems all the more improbably because Burr himself was a slaveower. [pp93]

It is hypocritical positions of Burr like slavery that caught the eye of Hamilton and other contemporaries who saw him as, essentially a hypocrite.

The truth, ultimately is that Burr is a complicated character, who was filled with contradictions. That made himself a target for political opponents like Alexander Hamilton who were quite vocal in their personal beliefs and attacked others who abstained from being clear on their own policies.

Can I add a follow-up question? How accurate is Gore Vidal's portrayal of Burr? I know it's a novel, but it's my first time wading into this time period in American history.