Well, at the risk of creating an 'alignment war'... evil is pretty much definitionally a lack of regard for the welfare of others. At the VERY LEAST this is a hallmark of evil. So, the characters participating in an 'evil campaign' are basically BY DEFINITION unconcerned with each other's welfare (beyond some sort of narrow self interest obviously). The classic evil campaign we ran for a few years pretty much worked on the premise that there was, at some level, an 'evil overlord'. If you screwed up the 'evil side' too much with whatever antics you were up to, then you got offed (because Mr Evil Overlord squashed you like a bug, or more like some of his sub-lieutenants did that). NO PC ever learned the identity of the evil overlord, and if you did rise to a level of power that seemed a bit too much, you found your 'usefulness' started to be doubted!
So, the basic equation was, you could backstab or whatever, but if it compromised the outcome of your current mission, then you were probably SOL unless you had a really good excuse, or could get something on your immediate superior so he wouldn't just off you. Thus the PCs often DID cooperate, and in many cases there wasn't a lot of reason for them NOT to do so. Albeit someone was always scheming and each character constantly either sought out some way to get more powerful than the others, or else to force them to work for him, etc. After a while some PCs rose into the ranks of the 'supervisors' a couple levels, and of course it was all just more of the same! Except now you had minions (who were often other PCs, this was after all AD&D).
Overall it worked pretty well, but there were certain inherent limits. Any PC that was weakened for some reason was either ripe to be ganked and looted, or put under another's thumb, so it was hard to survive and actually advance. Beyond that, the campaign could only go in a few limited directions or else it would 'blow up'. Without an evil overlord any pretense of cooperation would evaporate (we learned this). In fact the whole thing eventually crashed and burned because some certain player's character decided to help the good guys in order to get something or other, and evil's organization got mostly destroyed, after which the PCs just all murdered each other left and right. It was interesting, but it pretty quickly got old and futile.
I think it was a great success, lots of fun was had by all (many hilarious and stupid deaths of PCs) but it was not the most long-lasting possible format. I guess if you could come up with some sort of really credible ideology for evil, that might work better. Maybe something "We Orcs are the best!" or something.