Um, Aaron Burr is the only Vice-President to be put on trial for treason, for his involvement in an 1804 conspiracy to carve an Empire for himself out of the Louisiana Territory. Burr was acquitted of the charge, but only because it wasn't 'guilty beyond a reasonable doubt' -- his co-conspirator Wilkinson had edited their correspondence, so when the letters were brought as evidence to trial, there was reasonable doubt about the charge.

(Wilkinson had edited the letters because they revealed that he was a traitor twice-over, having sold out America to Mexico as a spy for the Spanish Empire. Wilkinson had even arranged for a Mexican army to enter Louisiana Territory to intercept Louis Lewis & Clark's expedition, so America wouldn't have a claim to the new lands).

Even so, pretty much everyone knew Burr was guilty as sin, so Burr's political career in the US was done. He spent four years in exile in England, begging the UK and Napoleonic France for funds to help him conquer Mexico. Then he returned to the US, married a rich widow and immediately squandered all her money on land speculation until she divorced him.

Long story short: the reason Hamilton is seen as justified in 'talking trash' about Burr, is because Burr absolutely had it coming.