I know Dave was in Metallica at first and wrote many of the songs on Kill 'em All and Ride the Lightning. The thing is, they were great, so why was he fired? I heard he had a bit of a drinking problem, but I don't think thats the main reason he was fired.
As you mentioned, Dave had a drinking problem. While drunk he sometimes had public outbursts or otherwise did things detrimental to the band's image. He also got into arguments with other band members and (according to himself) even got into a fistfight with Hetfield.
Check out this article from Ultimate Classic Rock for details: 32 Years Ago: Dave Mustaine Fired From Metallica (April 2015)
In Mustaine’s own book, Mustaine: A Heavy Metal Memoir, he attributes the final break to a physical altercation that took place between himself and guitarist James Hetfield after Hetfield got between Mustaine and original Metallica bassist Ron McGovney — and ended when, according to Mustaine, “I threw a right cross that landed flush against James’ face, turning his mouth into a pile of bloody Chiclets.”