Should i allow my PC's to...
My immediate answer to this is that, frankly, it's not your place as DM to allow or not allow a PC to
attempt anything the player wants. As DM you get to control a whole world's worth of NPCs, not to mention every monster the PCs face, and indeed the world itself. The player gets to control
one thing in the game, and that is his PC - get your hands off!
(There is a caveat to this, of course: cases of mind-control or other compulsion.)
That said...
Should i allow my PC's to kill another PC just because "he wasn't here"?
IMO it's not unreasonable to expect players to play their characters in good faith. At the very least, this can be taken to mean that they don't pull stupid metagame crap like this. The
player knows that that other player isn't here, but the
character should not. So in all but the most extreme of cases it would be wildly out of character for a person to randomly kill a friend (or at least a colleague) for no reason.
And, actually, that extends to
almost any intra-party fighting that resorts to lethal weapons. We are talking about groups of people who routinely venture into dangerous lands and fight unknown perils. That's just not something you do voluntarily unless you can trust the other person implicitly.
An intraparty squabble that devolves to daggers drawn (or magic equivalent) should be vanishingly rare. If things even look like they're heading that way, the adventuring group is vastly more likely to simply go their separate ways.
So, a player who declares his PC is going to attack and attempt to kill another PC without
extremely good reason is probably engaging in problem play. Done repeatedly, he should probably be ejected from the group.
(Two final caveats to that: firstly, this assumes a largely-cooperative game like D&D. Some other games, or some other campaigns, will have different assumptions and so may well have different expectations for sudden betrayals and the like. Although even there there should probably be
some sort of motive for a lethal intra-party fight. And secondly, of course, there are always exceptions - if your group has agreed that the PCs are just random psychopaths then have at it!)