Aaron Burr was hardly apolitical, he being both a Senator and Vice President at one time as well as a leading Gubernatorial candidate for New York in 1804. Indeed, the establishment of Tammany Hall and the New York political machine can be credited to Burr, who would deliver the state's twelve electoral votes to Jefferson in 1800.
That aside, Burr's political leanings somewhat reflected the core values of the Democratic-Republican party (especially his affinity for the west), though he would prove considerably more moderate than Jefferson on most issues (and having a collection of Federalist allies). I don't believe it would be inaccurate to liken him to Madison or Monroe (though Burr would have proven much more of a loose constructionist). Were he to lead his own political faction (rather than take the reigns of leadership over the D-R's in the wake of Jefferson), it would lie somewhere between the Federalists and their opponents.