The gospels present two different examples of guys who betrayed Jesus.
The one we all are familiar with is Judas. The other is Peter the Apostle.
Judas I understand, repented in the sense that he realised the awful thing he'd done, but too late. He also thought the Pharisees would only arrest Jesus, but not crucify Him. When he realise what he'd done he was overcome with remorse. That would have been fine, God is forgiving (and as Jesus said "forgive 77 times 7 times" in other words, A LOT) and would (my guess) have accepted Judas' repentence despite the awful thing he'd done (which is good news for us sinners!). But Judas took it step further. Not satisfied with asking God's forgiveness, he went and hung himself. Now I know this mightn't go down too well with some, but in Christian theology, what Judas did was to give in to "the sin of despair"
I'm not talking of the kkind of desperation you feel when the whole world is against you, but of the belief that you are beyond God's redemption, so evil are you. this is one of the devil's tricks: make you believe you are so evil you cannot even approach God for forgiveness. Thus you turn away from Him (and salvation) of your own free will. There is an early example of this in Gensis, after Adam and Eve had eaten the apple (which Satan had advised the to do in direct contradiction of God's law - btw the symbolism of all that is too much to go into here). God went to look for them and found they'd hidden themselves away. He asked "why are you hiding from me?" and they answered "because we were naked" (perhaps, in other words, they became self-aware for the first time). Now God's answer is VERY interesting. He does not say "why are you hiding because you're naked" as you might expect but "WHO TOLD YOU you were naked?" (Genesis 3: 8-12)
I find that sentence fascinating, because it indicates to me that (as God knew) the idea that we are naked and must hide from God in our shame came from an outside, evil source that wanted to keep us away from God.
And that's the situation I suspect Judas found himself in. Ashamed, he avoided God and took his own life (which only God has the right to take - hence the 5th commandment).
So he went on a path away from God and continued on this path after death. Thus, though he repented he also believed himself beyond redemption and spurned God (though not in a snobbish way, but in an inverted snobbish way, if you get the idea).
Satan hates us with a vengence - because we, unlike him, can be close to God, and because God loves us. Satan would love us to destroy ourselves.
Judas' sin was against God, so only God could forgive him, and Judas' did not accept that forgiveness which is part of the deal). It's a bit of a red herring to ask Christians to 'forgive' Judas for the same reason, plus someone cannot be 'forgiven' posthumously, at least not in the Christian sense - that kind of forgiveness is only possible in this life (unlike civil 'pardons'). We can forgive the sins of those who sin against us, basically by wiping the slate clean for that person and asking God not to hold it against them on our account.
Peter on the other hand sinned not once, but three times by denying he knew Jesus, as he was afraid of being crucified himself (which should also be good news for us, as it shows us that even Jesus' "number 1 apostle" was a weak man, but made strong through God). Jesus knew Peter would do this "before the cock crows, you will have denied me 3 times" Never! said Peter, but it was so. Peter was overcome with disgust at himself abandoning his long time freind and leader. He went out and like Judas, wept bitterly. But unlike Judas he did not kill himself but sought God's forgiveness and went on to be the cornerstone of the new Christian church.
So those two examples even tell us about our relationship with God.
OK, I know I'm speaking Greek here to some people, and I'm no expert, but the question was posed, and since the Christian act of forgiveness is being mentioned, it can only be explained / answered from a Christian perspective. I have tried to do so as best I can from my understanding and I hope the answer is satisfactory. I reject all attempts to be accused of "circular thinking" since I am trying to answer a question from within the framework from which it was posed ;-))