We know that he gotten taken in by a thieves' guild at age 14, and he was chasing Regis up north about 15 years later. We can safely assume that guild life in the intervening years taught him his tricks re: stealth, but where did he learn to kick ass?
Drizzt trained intensively with Zaknafein for 3 years, then went to the Academy for another 10 (IIRC). He had close to 15 years of hard training under the tutelage of confirmed masters to hone his talents and learn his trade. What about Entreri, though?
Given how formidable his reputation is at the point where we meet him, we can assume that being a badass is nothing new for Entreri. He's been the baddest man in the whole damn town for many years. As such, the most likely scenario to me seems to be that--relatively early in his tenure as a teen in the Basadoni Cabal--Entreri must have sought out a master or masters to train with in his free hours, maybe working his way through several as he quickly exceeded each.
I can't recall that it's ever been addressed; seems like there might be a story worth exploring in there somewhere.