Love keeps no record of wrongs

Love... keeps no record of wrongs (1 Cor 13:5)

Surely that must be one of the hardest things in the bible. But it's there. This is the love that God has for his friends and his enemies. And it's the love that he wants us to have for our friends and enemies. So part of forgiveness means forgetting the hurt that has been caused us by someone.
Don't get me wrong, I'm not suggesting we should act as if we hadn't been hurt when we have been:

Bear with each other and forgive whatever grievances you may have against one another. Forgive as the Lord forgave you. (Col 3:13)

When we're wronged by someone, it might be that it doesn't bother us and so we can just forget about it straight away. I take it that's what Paul means by bearing with one another. At other times when we're wronged, we also genuinely hurt by the person - sometimes to the point where the consequences may last for months, years or the rest of our lives. When we're hurt, we're right to tell the person that they've hurt us (see Eph 4:15), but that we have forgiven them - or that we want to forgive them. If we're to forgive people in the way that we have been forgiven, then that means forgiving people even when they don't repent or, worse, don't think they've done anything wrong:

Jesus said, "Father, forgive them, for they know not what they do." And they cast lots to divide his garments. (Lk 23:34)

Guiding principles
But it surely won't always be right to act as if some wrongdoing never happened. If a beggar asks for money for food and you then watch him use it to buy alcohol, it might not be the most loving thing to give him more money the next time he asks. If someone mugs you, it might not be loving your neighbour if you decide not to report it to the police (since your neighbour might be the next one to get mugged!). So we might ask ourselves "how can I be most loving in this situation?"
Also, we are not God and as a consequence, have limited resources (time, money, energy, etc.):

Conduct yourselves wisely toward outsiders, making the best use of the time. (Col 4:5)

Having said that, we need to balance this against our natural desire to keep our resources for ourselves!

If someone wants to sue you and take your tunic, let him have your cloak as well. (Mt 5:40)

Surrendering the right to revenge
Forgive me(!) I've already gone on too long. Let me finish with what I have found most helpful when trying to forgive others (which I learnt from John Piper's book "Future Grace")...
There is a sense in which we have a right to seek revenge (in the sense of exacting justice) when someone wrongs us:

If anyone injures his neighbor, as he has done it shall be done to him, fracture for fracture, eye for eye, tooth for tooth; whatever injury he has given a person shall be given to him. (Lev 24:19)

But Jesus calls his followers to give up that right:

"You have heard that it was said, 'An eye for an eye and a tooth for a tooth.' But I say to you, Do not resist the one who is evil. But if anyone slaps you on the right cheek, turn to him the other also. (Mt 5:38-39)

How will we ever be able to find the strength and the grace to do that? By trusting that God will settle accounts on judgement day. The person who has wronged you will either face God's punishment for what they have done; or God will have already punished their wrongdoing in the cross of Christ.

If it is possible, as far as it depends on you, live at peace with everyone.
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.
On the contrary: If your enemy is hungry, feed him; if he is thirsty, give him something to drink. (Rom 12:18-20)