I've been a player in an all-evil campaign, and the only thing we struggled with was ongoing Motivation...
Mostly, as players we were all really into it, as an experiment on a different kind of play style. No-one actually got especially "evil" in terms of behaviour, more just various different takes on "being evil". For example, one player took the classic "BBEG" role, tried to treat the rest of us like his henchmen, and it was great fun because we were all long-time mates and took it in good humour. We also had some classic moments where one or more of us would couple deception with random action, and all of a sudden talking to an NPC turned to cold-blooded murder (and then it went to "oh crap, we've got to hide the evidence!"). Which isn't neccesarily much different from the typical "murder hobo" style some people play with, but for me (player or DM), that's not our modus operandi. Certainly, we resorted to killing as our first choice, more often than not, as long as we thought we could get away with it.
Where the game actually fell down was with the DM... he struggled to transition from the typical D&D game, to providing ongoing motivations for our PC's to actually do anything much except randomly wander around a town then dungeons etc. So the game collapsed, because our DM felt he couldn't come up with decent motivations for us adventuring. Which was a shame, but yes he was struggling, he seemed to expect a typical D&D 'hero' campaign to be just as compelling for the 'bad guys'. In other words, he hadn't really thought it through, about what an all-evil campaign might need to be like, to succeed.
So... personally I'd suggest that you need two things:
1) 100% mature people who are invested in a good experiment
2) a DM who can help shape the campaign, with the players, providing good motivations for 'doing stuff'
For a one-off adventure, something as simple as "play the bad-guys" could easily be fun. For example: guard your dungeon from invading Heroes; assassinate the Heroes before they get to your Boss; stuff like that that flips the usual D&D trope on its head.
For an ongoing campaign, treat it like any other campaign - give your players a reason for their PC's to stick together (e.g. a common a Boss, Faction, Guild or Goal that they all share), then take note of their background and goals, and structure adventures that work with those. Evil isn't stupid, it just goes about its business in a different way.
So yes, being Evil might make the PC's jobs harder, and lead to more complications, but if you have mature invested players, there's no more reason for it to turn to back-stabbing as with any other campaign. The added level of complications, and hence Challenge, is exactly what mature people find attractive in an all-Evil campaign.