The death penalty is wrong for many reasons. If someone is innocent, but found guilty, you can't go down to the graveyard and dig them up, dust them down, say sorry and send them on their way. If someone innocent is killed like that, shouldn't the logic of the justice system then say that the judge, jury and executioner should be sentenced to death for killing an innocent person? Do supporters of the death penalty think that should happen? Have you ever even heard of it happening? If you do support the death penalty then that is exactly what should happen.
There are the contradictions of the death penalty. The fundamental one is the justice system saying "You murdered someone, murder is wrong, so we are going to murder you." You can't have it both ways. If it is wrong for the person to kill someone, then it is wrong for them to be killed. You are effectively condoning what they are doing. There is also the one I already mentioned, the contradiction of anyone from the justice system not facing the death penalty if an innocent person is executed as a result of their actions.
Of course, if you do support the death penalty then logically you would never use it. That sounds stupid you are saying, but think about it. If you support the death penalty, then you think it is ok to kill people, so if someone is found guilty of murder you would not see them as having done anything wrong, so you'd see no reason to punish them at all.
If we murder murderers, shouldn't that mean that we should rape rapists, sell drugs to drug dealers, steal things from the homes of burglars and so on? We'd have no need for prisons at all, well maybe with the exception of someone being found guilty of imprisoning someone.
If someone is guilty of murder or other very serious crimes, then lock them up permanently, or at least for a very long time.