I've run long-running evil campaigns that worked very effectively. What worked for us was to begin with these two ground rules in mind.
Forget the word "evil". The in-character concept goes like this:
People who do not have the willpower to effect their will often judge the highly motivated as evil. We do not think of ourselves as "evil." We are decisive; dedicated; intent on achieving our goals. In fact, we often serve high principles and the common good--by any means necessary.
Which brings us to:
The Ends Justify the Means. Whatever it takes to achieve our goals, we will do.
Say we need info about a merchant's operations. We might bribe a shop clerk for the info. Or, we might kidnap the innocent fellow and squeeze him til he talks. Which approach is taken depends on how much time effort and/or money the characters want to put into the process.
At the end of the day, though, we achieve our goal by doing "whatever it takes."
Those who are more chaotic in orientation might do this with little concern for consequences, because Poor Impulse Control or Judgment. So the whim of a moment might dictate their actions. In the encounter with the kidnapped shop clerk, one might kill the prisoner after interrogation because "I didn't like how he looked at me."
Those more lawful in orientation will act with cognizance of the hierarchy and expectations upon them and any 'rules of the road' their leadership has asserted. "We let him go after interrogation because the Boss wanted to see who he'd run crying to."
In either case, the strategic End justifies the tactical Means.
If characters do this in self-serving ways that disregard norms or what others want, their actions can easily be called evil. This does not mean they are not capable of acts of kindness, love, caring for family, friends, or even trusting each other. Those elements can still be in place. Their "evilness" arises from how others perceive their actions. Not (as others here have noted) because they are cackling in a corner, rubbing their hands, and saying "Let's be EVIL! Bwahahaha!" (Although, ok, some players might be doing that. LOL)
Final note: setting expectations
I always had an "expectations-setting" discussion/meeting that was its own separate thing for all the players as part of the required preparation for this kind of campaign. The ground rules above were outlined, behavioral questions and motivations explored. Individuals and group were encouraged to come up with some goals they would strive for, in the course of which they could explore their 'evilness'. We'd check in periodically, too, to make sure of people's comfort zones and any other dynamics-related things that came up in game play.
This worked out really well. Occasionally there'd be a "drunken frat boy" style player. I would have private convo with him to see if I could get him on the right page for this style game play. If he remained randomly disruptive, he was uninvited from the group. Only had to do that twice over several years.