So why isn't it murder when the government does it?
Merriam-Webster defines murder as "the unlawful premeditated killing of one human being by another." The death penalty is indeed premeditated, and it is indeed the killing of a human person. These two facts are indisputable. But it's lawful, and it's not the only example of the lawful, premeditated killing of a human person.
Many military actions, for example, falls into this category. We send soldiers out to kill, but most of us don't call them murderers--even when the killing is part of a strategic attack, and not a form of self-defense. The killings that soldiers perform in the line of duty are classified as human kills, but they are not classified as murder.
Why is that? Because the majority of us have agreed to give the government conditional power to kill with our permission. We elect the civilian leaders who order executions and create the conditions for military killings. This means that we can hold no single person or identifiable group of persons responsible for such deaths--we are all, in a sense, accomplices.
Maybe we should consider the death penalty murder--but murder, like all crimes, is a breach of social code, a breach of the rules upon which our society has more or less agreed. As long as we elect civilian representatives to impose the death penalty, it's very difficult for us to say that it constitutes murder in any commonly used sense of the word.