I don’t believe that Jesus Christ approves of the death penalty. I think His death on the cross changed the “eye for an eye” mentality of the Old Testament days. Jesus explained this when He dealt with the woman about to be stoned for adultery.
Government-sanctioned killing and murdering of the mentally ill and the poorest in society are not Christ-like behaviors. These people are often minorities who can’t even afford a good attorney. Killing those who kill is not logical. We have made ourselves into the murderer.
As Christians, we have to accept a turning from the need to avenge the wrongs done to us; we are to turn to God who will handle wrongs that happen to us in His time. I’m often surprised at the bloodlust many Christians express as “godly.” Cheering on death for another seems very anti-Christ to me.