Burr shot and killed Hamilton who was a likely candidate for president in 1804.
Not quite. Hamilton, though very successful as first Secretary of the Treasury under Washington, was never
a likely candidate for President. In fact, only Burr had a realistic shot at national (or even state-wide) office.
Now it is very likely that presidential politics played an indirect role in the duel --since Hamilton tried hard to insure that Burr did not
become President in 1800-- but the more immediate cause was to be found in NEW YORK politics. Hamilton had made some sort of vicious tales about fellow New Yorker Burr while the latter was running for Governor. Burr heard reports of this shortly after losing that race, and asked Hamilton to specify what he had said and/or deny; Hamilton, as you can guess, never did. (Historians to this day are trying to guess what Hamilton said --best guess I've heard, and still just a guess, had suggested an improper relationship between Burr and his devoted daughter). Of course, it's likely Hamilton really could not deny having said such things, since he had been attacking Hamilton for years. (It may be that Burr had decided he just wouldn't take it any more, or that the charges in this case were particularly serious [which is why some are attracted to the theory that it touched the honor of Burr's daughter].)
That was the immediate cause of the 12th amendment that did away with the original system for choosing VP's.
The immediate cause of this amendment was the election of 1800 in which Jefferson & Burr (the latter understood to be the V.P. candidate) ended up in an unintended tie in electoral votes. Since everyone was aware of the danger of a tie, the plan was for a couple of electors (yet not enough to endanger the success of the ticket) to cast their ballots for a dark horse to ensure that the intended candidate ended up in the top spot. But in 1796--the first time Jefferson & Burr ran together-- southern electors had failed to give full support to the Burr half of the ticket, angering the northern wing of the party. Thus in 1800 the effort was made to avoid a repeat of that mistake (especially after Burr's efforts successfully lined up essential support for the ticket in New York), but they did too good a job of it!
Following the original Constutional provision, the tie was to be settled by the outgoing House. This made matters especially messy, because though Jefferson's (Democratic) Republican party had won control of the House, the outgoing body was heavily Federalist and very much against Jefferson. Burr was, of course, of the same party as Jefferson, but a number of Federalists did not believe he was as partisan. Feeling they could work better with Burr than with Jefferson, some voted for him. In a series of inconclusive votes in the House, Burr actually stayed very much on the sidelines, rather than courting Federalist votes. Had he done so, he likely could have secured them. (Ironically, Jefferson, who did court votes, was led to believe Burr was trying to do just that, never trusted him again, and eventually dumped him from the 1804 ticket.)
Anyway, to be fair to the Founding Fathers, when the Constitution was written everyone at the time despised
the idea of political "parties". Having the second-choice (hypothetically the second most qualified) as V.P. made sense. With the emergence of the party of Jefferson & Madison (called "Republicans" at that time) in opposition to the Federalists (Washington, Adams, Hamilton...) it became apparent that idea would no longer work.