If a child committed murder, how would it be punished?

Most likely the child would be put in a juvenille facility, although it really depends on the laws of the state/country where the crime took place as well as the circumstances and the leaning of the individual judges in such cases.

I have heard of juvenile murder cases fitting the parameters you have described being tried as an adult murder case, but those are the really exceptional ones.

To answer the unasked question, yes, children have been given capital punishment for crime in the past, although today it is illegal in the United States, and I quote from Wikipedia on this: "The United States Supreme Court abolished capital punishment for offenders under the age of 16 in Thompson v. Oklahoma (1988), and for all juveniles in Roper v. Simmons (2005). In addition, in 2002, the United States Supreme Court declared unconstitutional the execution of individuals with mental retardation, in Atkins v. Virginia."

In the US, there would be no capital punishment. If the child were convicted, they would probably spend the years until they are at least 18 and most likely 21 in a juvenile facility. At that point, if they are believed to be rehabilitated and not a threat to society, they would most likely be let go.

The laws are slowly changing though, to allow the authorities to hold a person they believe poses a risk to society until they are no longer a risk.

Nothing really. He would go through some testing,find out if he's mentally stable. Of course if a child kills somebody my guess is they aren't. In that case that child will be put in a home for the rest of his life.

probably locked up at the age in a juvenile facility so he can learn to do other crimes and not get caught

Appropriately, hopefully. It's clear that you've ignored the many stories of children who've killed.