With respect to the "eye for an eye" topic, such instructions given in the Hebrew scriptures were intended to limit retaliation, not encourage it. If someone poked your eye out, you could not then poke out both of his eyes. Or if someone broke your arm, you could not then break his arm and his leg. And then Jesus came along and turned even that upside down. In short, "turning the other cheek" was a means by which the persecuted could nonviolently demand equality.
But with respect to the death penalty/capital punishment and, along the same line, prisons...
I believe that God desires more than punitive justice. If we are called, as His people, to bring His kingdom to earth, then what place do prisons and the death penalty have? The U.S. has the largest prison-industrial complex in the history of civilization. In some ways, imprisonment is an alternative form of slavery: corporations can hire inmates and pay them less than minimum wage, all the while denying them the chance to experience redemption. Is this what the Kingdom of God looks like?
But beyond that... If we seek to follow Jesus, how can we rejoice in (or advocate for) the death of anyone?
It is a common attitude to think "Well, what about people who will never repent or have committed the most heinous crimes?" To that I would say "Are we saying that some people are beyond redemption? And if some people are beyond redemption, then what does that say about grace, reconciliation, and the power of Christ?" Much of the Bible was written by murderers who were given a second chance (David, Paul). Redemptive violence is a lie.
Jesus was directly questioned regarding the death penalty when He encountered a crowd ready to stone an adulterous woman. True: her crime was capital, and her execution was legal. But just because something is legal does not make it right.
The early Christians were characterized by nonviolence; but that changed under Constantine. And that period gave rise to the violent military complex that dominates today and that is largely supported by professing Christians. How horribly ironic that we drop bombs and ignite electric chairs paid for with money that says "In God We Trust"! To those who see such contradictions, especially those who do not follow Christ, I wish we could say "This is not my Jesus."
Honestly, I used to support the death penalty...along with the military, exploitative economic policies, and some other things I no longer condone. These convictions are relatively new for me, but I hold them deeply.