PvP is always touchy issue.
One of the basic rules of playing RPGs is that you should never try to deal with out of game problems with in game solutions, and never try to deal with in game problems with out of game solutions.
So one of the first things I'd try to do here is figure out what are the in game problems and what are the out of game problems.
The problem you usually see in PvP is that a player is frustrated with another player, and he decides to deal with that problem by having his character take out his frustration on that other player's character. The other problem you tend to have is that even if that isn't the motivation, having a conflict between two characters tends to create frustration between two players.
Out of game problems need to be talked out, usually resulting in some sort of social contract. In game problems need to be role played out, preferably guided by the social contract and mutual respect between players and not just what they think the character would do.
As a general rule, I'd advise most groups of new players to have the following explicitly in the social contract:
a) You must play a character that has a motivation to stay with the group.
b) You must play a character that has a motivation to be allied with the group, and sees the success of the group as critical to their own success, regardless of what that character's underlying motive happens to be.
c) You must play a character that won't immediately be drawn into conflict with the group, or which at the very least won't as a first chose attempt to resolve conflicts violently.
As a DM, I'd be looking at this table conflict with questions like:
a) Is someone being annoying out of character, and is someone letting themselves be annoyed by that?
b) Are people choosing to play characters that fit with the group, or are they deliberately trying to draw spotlight to themselves by being 'the outsider'?
c) Is the player's play sophisticated and mature enough to be the outsider - the Belkar Bitterleaf - in a group? (In this case, clearly not.)
d) Does the player have story goals in mind and is he signaling ways to work out the problems in a way that everyone has fun? If the players are signaling that they want a mutually fun solution, then let them play. But if someone is just being pig headed about the fact that their character would leap to murder a comrade at the first opportunity, probably we've got a problem with the social contract and we need to work out what everyone wants from the game (or at the very least, what the DM wants from the game).
As a group matures, you can have more sophisticated plot lines with betrayals and conflicts and that sort of thing, but those sort of plots when they work well involve players who can engage in sophisticated signaling of consent and purpose and who are motivated by the mutual sense that "this is a really cool story" and want the story to stay cool. Pulling off a betrayal or conflict plot for purely selfish reasons is always a bad thing, because fundamentally RPing is a social activity where everyone is trying to have fun. One of the earliest groups I was involved in though their main campaign, after obtaining name level, they'd consensually allowed one of the PC's in the party to become the BBEG of the setting, and the main storyline revolved around the remainder of the party dodging the attempts of the PC to kill his comrades while they organized a rebellion against him. That's pretty cool, but not every group can pull that off well, especially without fully relegating the evil PC to NPC status.
As a player, I have been involved in these scenarios several times. It's my experiences in 30 years of gaming that 80% of people can basically only play themselves. Well, in one group there was a newbie who could only play Chaotic Evil characters. Regardless of the character he conceived, the character would be basically Chaotic Evil in conduct. And there was also a newbie in the group who could only play Lawful Good characters. Regardless of the character he built, the character would be scrupulously honest. There was no point in putting skill points in 'bluff' because it would never occur to the player to be dishonest. There was really no point in writing anything but lawful good on the sheet because his first instincts were always to be heroic, valiant, fair, and honorable. The only time I ever got him to compromise was when I could get him to have conflicting duties to multiple groups, in which case he'd be perfectly lawful good and organize his loyalties and duties into a numbered hierarchy and decide which had priority in this situation.
Needless to say, despite being friends, in play these two players were continually at loggerheads. During one campaign, the LG player was playing a LG character and the CE player was playing a CE character, and the LG character discovered the truly vile IC activities the CE player was engaged in. To preempt the LG character bring the CE character to justice (no vigilantism here, remember), the player decided his CE character would kill the LG character. The two mortally wounded each other, and then I had my character act to let both of them die of their wounds and prevent either from receiving medical attention because that's what my character would do in the situation. And that was the last of the PnP we had. (We probably wouldn't have had it in the first place if I was the DM, because I wouldn't have approved the CE character for play.)
But I've also had PnP wreck campaigns, with parties basically having TPKs when things went to blows and the few survivors not being in a shape to then survive the scenario.