I think this is a typically difficult question to answer. Firstly, religion massively influences people's beliefs on this. But my belief is exempt of any religious bias. I was indoctrinated in to Catholic values until I reached the age of reason, and realised I never wished to be associated with any religion at all (including atheism). So this is purely based on my own moral compass, my own abilities to decipher between right and wrong.
I believe in compassion for life in general (and that includes other terrestrial and extra terrestrial species). And Murderers, Rapists, etc. are a clear breach of any compassion for life. Once that boundary is crossed, these people deserve to be punished in some form. First I think it's reasonable to put myself in the victims loved ones shoes, for example, someone had raped and murdered my daughter.
What emotion would that evoke? Its all good trying to pretend here that I am all holy and would forgive everyone no matter the level of evil the person has to commit this act. Do I think forgiving someone like this is a noble thing to do? No I do not. Doing so is sheer denial of natural emotion.
I don't believe in violence either. But if someone did something this sick and abhorrent to a loved one of mine, my moral compass tells me I'd be right to kick the living shit out of the guy. Would it change the original crime from occuring? No. But that venomous release would be healthy. And justified in my opinion. And if that person was beaten to the point of death, I wouldn't see that person as a murderer. I'd see it as someone who loved his daughter.
There's external factors that affect the criminals behaviour such as mental illness (perhaps they're sociopathic or psychopathic) In which case, it's not the criminals fault. If a person isn't in control of their moral compass due to this, they need to be segregated from society totally.
Having a rough upbringing is a different kettle of fish totally. Criminals may have been manipulated, raped, beaten, etc. by their parents or guardians. Its horrible, and certainly a huge disadvantage in terms of knowing what's right and wrong. They weren't taught that by their parents. They were used to being the victim and their view of normal behaviour differs to the general concensus. As much as I agree that people like this are more likely to go to a life of crime, lots and lots of people have had the same kind of background and went on to harm no one (incredible amount of respect for these people by the way) so inflicting that same pain on others is not necessary.
Anyway, the state killing serious criminals after years of imprisonment (with their pool tables, TV, And being Bill-free) will not bring justice to the families nor prevent the crime from happening. I'd also like to point out that some people do change and are repentent. Some have left jail and vowed to help the sick or poor to try to make up for their crime. So killing them prevents the chance of people changing.
Some people could be helped by rehabilitation. It really depends which method suits which type of criminal. Of course, certain criminals are 'beyond repair'.
Sometimes Capital Punishment is carried out by flawed justice systems. E.g. racially aggravated, black people more likely to be put on death row in certain places and China disclosing the amount of people sentenced to death. I don't trust these authorities at all.
Statistics have shown that the Death Penalty hasn't really acted as an effective deterrent (really surprised me to be honest) it's also very expensive keeping convicts on death row, eating in to the tax payers (potentially innocent people) money.
For these reasons, I'm against Capital Punishment. And murdering criminals seems a pointless action given the fact the damage has already been done. I can however there are many counter arguments that I agree with. Which is why I don't defend the abolishment of Capital punishment as strongly as Amnesty for example. The ability to see both sides of a debate is a beautiful thing.