Okay, here's my elaboration. When Harvey Dent was D.A. he was essentially one of the good natural good guys. Everything he did was to ensure the safety of Gotham City from the bad guys. Now, it was the Two-Face aspect of himself that pushed him to actually start killing people while he was still Harvey Dent. That there brought him to being a vigilante. I feel that the vigilante aspect of Two-Face comes from both sides in a strange way working together. Being a vigilante is in itself a crime, especially when you're killing the criminals. In the works of Two-Face, the vigilante, Harvey is putting away the crime while Two-Face is making sure they are put away as ultimate executioner.
The crime boss aspect I feel is more completely the Two-Face (let's call him Murray based on Batman: Jekyll and Hyde) side. Murray is the evil aspect all the way. Without any restraints, without the coin and without the Harvey aspect, he would be a no-holds bar madman making sure everyone suffered. And one thing the Murray side loves is crime and being some sort of a terrorist. The Murray side always represented the things Harvey hated the most in society and Gotham, and now its what his "partner" glorifies in, which brings about the crime boss aspect. We know how much Harvey hated crime bosses and the mob, etc, and now he becomes what he hates the most, which adds to more of the tragedy of the character when he has to take a sad backseat and see the wreakage he's causing. Now, just like how the Murray side causes Harvey to kill as a vigilante, the Harvey side should find some way to make Murray do good as a crime boss, but we don't see much of that. I feel that's something that should be elaborated and worked on more by writers. The Harvey side that does come out, though, is the side that allows certain characters to live and allow Murray to sit back and allow him to help someone in need.
So through those is how the two combined together really allows the character of Two-Face to be both a vigilante and crime boss and still don't seem out of character, at least for me. The coin helps him do all types of things. I do feel, though, that we don't get to see too much of the vigilante side. That I would love to see. The one thing I do find scary about the character though is in portrayals like David Hine's Two-Face one-shot in Joker's Asylum. It seems everything being done is due to the Murray side, but it seems as if it's Harvey also trying to showcase just how cruel life is, which sorta mixes both sides and aspects of the character and makes him even more dangerous. All I know is, I love the different interpretations of the character... except for the Batman Forever interp. Yeesh.