Explain how the duel between Burr and Hamilton relates to the issue of character

In 1804, after Burr had been defeated in the New York Gubernatorial race, he learned Hamilton had allegedly made some disparaging remarks about Burr's character. The particulars are unknown because Hamilton's insults were only implied, not quoted.

A man named Dr. Cooper wrote a private letter to an acquaintance, and the letter was somehow leaked to a newspaper. The part Burr found offensive:

"Genl. Hamilton and Judge Kent have declared in substance that they looked upon Mr. Burr to be a dangerous man, and one who ought not to be trusted with the reins of Government. I could detail to you a still more despicable opinion which General Hamilton has expressed of Mr. Burr."

Burr exchanged several letters with Alexander Hamilton requesting an apology, but Hamilton refused. Burr's initial approach seems to have been relatively straightforward and non-confrontational, but Hamilton's response was taunting and Burr's anger eventually escalated to the point of challenging Hamilton to a duel to defend his honor.

Hamilton accepted, but history tells us he intentionally fired into a tree limb about fourteen feet above and four feet wide of Burr's head. Burr either believed Hamilton intended to shoot him or simply took advantage of an opportunity (the truth has never been determined), and fired directly at Hamilton, hitting him in the abdomen. Hamilton sustained organ damage and a severed spinal cord, and died the next day, July 12, 1804.

Burr was charged with murder in both New York and New Jersey, where the duel was held, but the case was never prosecuted. So certainly the issues of character and slander play heavily in this duel. Did Hamilton fire above Burr on purpose or was he just a really bad shot? How does this define his character? All of these questions have been addressed with this incident.