Who decides which people need to be killed? What is it that makes them very bad? What makes you want to kill them?
Is this a pre-existing, identifiable need to be exterminated thing? Or more of an at the moment their badness manifests sort of thing?
You're not seriously asking that, are you?
In the larger scheme of things society does this through the justice system. That's what laws, courts, police, etc. are for in that larger context. Each society has its own answer to the extent it will go in this area. And it is of considerable debate.
In the context of war it is generally fairly straightforward -- the soldiers of the other side. It can get very messy, of course, but that is a discussion for a different forum.
In the context of someone breaking into my house and actively approaching my family armed with a large kitchen knife, well, the person who decides this issue would be, well, me in my hypothetical. I would be deciding to protect myself, my family, or in other scenarios someone else for that matter from a person actively trying to do harm. Certainly not an easy decision to make and I've never had to make it. And I don't know how I'd act under that kind of stress. However, I do train with some awareness that should something like that happen I hope to be able to act decisively. And likely then fall apart later like a little schoolgirl.
Did you have some other scenario in mind? Or is this conversation going to change from the OP's original question to the larger question of "who decides" as some sort of philosophical issue? Frankly I wasn't considering any scenario other than the last one -- the horrible situation where there is a life threatening event occurring. And I have a hard time believing that most people here were talking about anything beyond that a la a Dexter sociopath/psychopath or vigilante justice.