I have played "PCs as villains" in various ways. The more you want a long, traditional campaign play with all the PCs "in the group," the more constrained you will be in options - a one shot or a planned several session adventure, you can accomplish this all sorts of wild ways.
Covert Bad Guy In The Group
In a long Night Below campaign (AD&D 2e), I had one PC who was the agent of an evil organization (the Scarlet Brotherhood) which however wasn't the primary opposition for much of the campaign. He was urged to just "observe and report" and they'd feed intel back to him. The SB had its own agenda re: the other power blocs the PCs were interacting with. This could be modified easily for the PC Villain to be more proactive, but you need a similarly indirect goal. If the problem can be solved with "Well I kill these other three dumbasses in their sleep" then the campaign won't last long. You have to have some active reason for the PC Heroes and PC Villain, if they are hanging out, to not kill each other if the villain's "discovered."
Also note this model is fragile - that player eventually left the group. Since he was a subplot that was fine, if he was "the" villain that would have been harder (though entertainingly, the other players never knew he was a spy, so even after he retired from the party after becoming an NPC they would go find him to give him updates and ask his advice.)
Not So Covert Bad Guy In The Group?
This is harder, but you might be able to have an open baddie in the group under controlled conditions. It doesn't have to be a symmetrical relationship, for example, the movie Se7en comes to mind, where the two cops have the serial killer in custody for a good bit of the movie but his plan is already unfolding, so they're dragging him along in handcuffs trying to stop it. This is probably the only way to do long campaign, same group, with planned plot. Or maybe the "villain" is an evil artifact they are taking/using (Stormbringer!) that the plot relies on. But here you need a big, big, super overwhelming issue to keep the guy alive because it'll be very easy for, under pressure, them to decide at any given time "just kill him!"
Hard Hitting PC On PC Action
In one high level Forgotten Realms campaign (AD&D 2e) where we were playing Four from Cormyr, half of one party got vampirized into daywalking vampires and then they were all out in this ruin in a swamp with half the group trying to vamp the other half, and the others trying to kill and rez the vamps. It happened by accident but ended up spanning three play sessions, as they were equal in power and really got into the challenge of it. I had to go back and forth between rooms though, because much of the juice of this setup comes from tactically outdoing the other group. This is clearly a more time limited option, but hey, every campaign doesn't have to be a Three Year Epic (tm).
Player Plays BBEGs
I haven't done this with one player playing the same villain every session, but with major or recurring villains I like to have a real player show up and play them - it makes climactic combats "fair" and super hardcore. The players definitely know why that other player is there and it does require a little note-passing, but then again I tend to discourage metagame talk at the table so it isn't too disruptive. Tends to not be the same villain, but gets you some of that PvP frisson.
Players With Different Goals
OK now, this sounds like it's maybe what you're going for - players with different motivations who then inevitably come into conflict. That's cool and awesome. The problem is, you can't really plan it. PCs are/should be dynamic people, so you can plant seeds but you can't make "that guy the villain." It doesn't work that way.
In that same Night Below campaign, there was an eventual party schism along the lines of the "let's learn crazy Cthulhian lore and maybe sacrifice a party member if we need to in order to power a ritual to keep the world safe" faction and the "we are good people and we'll try to overcome but even if we don't it doesn't justify betraying friends" faction. I did a lot to encourage this and generally knew where the players would fall on that divide, but by its very nature is uncontrolled and up to the players. And maybe the players just say "hey, I'm the villain, whatever happens I'll just keep with the original plan" but the longer the campaign, the more unsatisfying that becomes.
I was a player in a Planescape campaign where I was a bladeling and we went through this whole plot where I started to get dreams from the Lords of Blades or something, that wanted to be freed to kill all the gods and start over. The party's plan was in general to thwart this goal. Once we got down to the end, however, I decided that I saw no good reason for it to NOT happen and plenty of good reasons for it to, so I tried to let them out. Oddly, this was counter to the DM's plans (aren't those the kinds of seeds you plant when you want this to happen?) and people didn't like the PvP that started, so I had to handwave giving in.
Now, there are various storygames, like Fiasco, designed to be played from less of an in character point of view and more from a playwright point of view, where you will all sit around and control the heroes and villains and therefore the need for information compartmentalization is gone. This is generally a major step away from the core kind of IC experience you get from traditional games, though, so you'd have to decide if you're into that or not. (I'm not.)