I am running a campaign in 4e, in which everyone is very very new, including myself. I have played a bit, but this is my first time DMing.
I have a Warlord PC who does not work with my overarching plans. His backstory also makes no sense as to how he would be level 1 — he's supposed to be a 40 year old veteran of war and the player insists it remain that way.
I've basically come to the conclusion that his character has to die. I have asked other friends who have played more, and they agreed with me. Their advice was to kill him mercilessly — like, he steps on a trap and is sent down a slide to fight a high level minotaur all by his lonesome.
My question is, as a new and hopefully long-lasting DM, how do I kill off a character? Do I make it fair? Is it better to instill the idea of hopelessness? Or am I overreacting and should let him play?
Short answer: You do not.
You say that he does not fit in with your plans as a DM. But the thing about being a DM is NOT that you tell a rigid story that your players walk through: instead you put them in a series of situations, see how they react and frantically try to fit your story to it. I understand that your story is your baby and the PCs all try to spike it to the floor like a bunch of murder-hobos would do with an orc baby, but that's how it goes.* One of the most difficult things as a DM is to let go of what you want to do and instead act on what the players do.
Not to say that there are no alternatives to your problem.
Talk to the player. Murdering a PC just because they "do not work with your overarching plans" is a downhill highway to getting a pissed off player. Instead, talk to them. Discuss this problem with them (though please LEAVE OUT THE MURDER PART) and see if you could reach some kind of compromise. What is this reason this Warlord is a 40 year vet and still only level 1? What is his actual story? Again, do not bring up killing them: the character is as much the player's baby as the story is yours.
Roll with it. One of the most interesting things as a DM is dealing with the bizarre stuff your players come up with. See what they want to achieve (especially the player in question), and try to match your story with this.
Talk to the players. The most difficult one. For the players, roleplaying is about experiencing a story, not being told one. They want to take part of it instead of being along for the ride like a rollercoaster. Ask them what kind of direction you want to take the game in, see how this matches your kind of story, and try to match it. If it works, great! If not, I'm afraid you'll have to compromise.
(* This is not to condone the murder of babies of any kind. Killing babies is a Bad Thing.)