How was Alexander Hamilton killed?
He was shot in a duel.
After a long dispute with Aaron Burr, he endorsed Thomas Jefferson, causing Jefferson to win the Presidency, and Burr becoming the vice. This angered Burr, because Jefferson and Hamilton had been enemies for a long time, constantly disagreeing. This led to a duel in Weehawken, New Jersey on the 11th of July, 1804, in which Hamilton shot purposefully above Burr's head, while Burr fatally shot Hamilton between the ribs. it is unknown if Burr purposefully aimed to kill, or if he was aiming to throw away his shot.
Source(s): "Alexander Hamilton" by Ron Chernow
Wikipedia: "Hamilton-Burr Duel"