I'm certain I'm reiterating someone at some point, in some of this, but I'll at least say I agree with most of the responses of evil campaigns not being preferable. I've run a few evil campaigns, to a Black Spiral game for Werewolf The Apocalypes, Super-villains in the DC Universe game from WEG, 3rd Ed D&D and have played in my fair share as well.
In my opinion, as far as games go, the only 'evil' campaign game out there that works is ShadowRun, since you're playing a member of a team of career criminals.
In most cases, it requires some of the utmost responsible and team minded players to play an evil campaign. In most cases, it's hard to find a regular 5 player group that can function like that.
In my own group, in the past, there has been the classic alpha male personality type who, in an attempt at evil campaigns in the past, he has the previously stated Anti-Paladin domination problem (and sometimes has that problem as a heroic character)
Currently there is a gentleman who is under the impression that if he hard-balls everything, they'll bend to his will (even a Great Wyrm blue who eventually was no-longer amused with the level 3 trying to hard-line negotiate with him and smushed him)
Those rare gems where the players all acknowledge "No dicking over other players" is the rare white buffalo, and a group that acknowledges that heroes, authorities, bounty hunters, small military contingents, will be coming after them and/or will attack them on sight within most cities are rarer.
Many times in evil campaigns, Cocky becomes a primary descriptor for the characters and their players. They feel that they can walk boldly into any town and all will tremble with fear and/or empty their coin purses at their very approach, and it quickly feels like Player vs GM when you make them wake up and deal with the consequences of their actions.
So like others stated, one of the best ways to run an evil campaign? A short one, maybe only a few sessions - and those short campaigns, maybe use them as a means to opening up a storyline for the regular campaign. Let the players help 'flesh out' the personality of some villains, and let them know that ultimately who those characters will be, later on, is determined by you, though take their own influences for the characters well in hand.
All in all, I play D&D to be heroic, (while one of my players has disturbed us by saying he plays D&D because he's not allowed (not unable) to do certain things in real life) If I'm running D&D, or other games really, it's to see a heroic narrative. If someone is in the group playing evil, rather than all, but he acts in a way that works with the others and aides their overall goals (while helping his own nefarious ones) so be it, as long as it doesn't turn into a series of party kills.