My personal beliefs, as a Christian, is that Jesus died for the sins of the world, and that upon our deaths, we are forgiven all of our sins and go to heaven. I do not subscribe to dispensationalist theology--the belief that only a select few are saved--nor do I believe that a single, or even multiple, sins are enough to damn people to hell for all eternity. Such would be unfitting for a God who "so loved the world that he gave his one and only son." (John 3:16 NIV). I also believe that people of all faiths are called to act with love and kindness towards each other, and while that does preclude murder, that also precludes the murder of the murderer (or, if you prefer, the killing of the killer). After all, that is what it boils down to... we condemn people to spend years in prison, simply waiting for it to be their turn to die. People seem to have this misconception that life in prison is some sort of a vacation. I believe that they are confusing prison with Holiday Inn. Watch Shawshank Redemption, and pay attention to the first half, before Andy makes friends with Morgan Freeman's character... prison is hell. Even after he makes friends, he is still miserable and unhappy. I think that life in prison, where you will have to think about what you did, and do so for a very long time, is a far worse punishment than death.
As for the argument of vengeance, "Do not take revenge, my friends, but leave room for God's wrath, for it is written: 'It is mine to avenge; I will repay,' says the Lord." (Romans 12:19 NIV) So... don't take revenge. Let God do that when/if He decides to.
Lastly, I think that capital punishment speaks to a larger problem in modern America, that of hopelessness. I think that, with regards to our criminal justice system, there is a great deal of hopelessness and despair regarding convicted felons. There is this belief that criminals are, somehow, intrinsically, internally bad, and therefore, irredeemable. The death penalty is the ultimate example of this. It is a statement, in effect, that "we think this person is so bad, so evil and twisted, that we don't think we should invest any time, effort or money in working with him to redeem him." I very carefully left out the words "without parole" in my statement above. The very existence of the concept is another example of this hopelessness, albeit a less severe and more reasonable one. I am not denying that there are people who will not repent for what they did, and who do pose a danger to society and will do so for the rest of their days. However, we need to have a justice system that makes allowances for people redeeming themselves and repenting for their crimes. Tookie Williams is the most prominent example I can think of in recent times, although I'm sure if I thought and researched some more, I could come up with many more. In reading about him and his work, I personally think he had reformed and should have been spared, but I know there are others who would disagree. It is also debatable if he would have reformed so drastically if he had not been sentenced to die, and that is a question I have no good answer to.
In case it matters, I would class myself as slightly Socialist, but mostly Democrat, and I am a practicing Episcopalian.