The holy rebbe of Piaseczna gives us a daring piece of advice in his seminal work Chovat HaTalmidim. At the end of the ninth chapter, the rebbe addresses the issue of reconciling with an enemy with whom the mutual hate is so great that you simply cannot find any redeeming qualities in this person.
This is what you should do. Write him a letter. Don't send it to him; hide it somewhere in your home. In the letter, insult and shame him as much as the serpent of anger in your heart desires. For some days, read the letter aloud, and imagine that you are standing in front of him, taunting and abusing him with all the expressions of the letter. After some days, you will find your anger has dissipated, and if you are a sensitive person, you may discover yourself running to reconcile with him. *
More than an act of catharsis, the rebbe's advice shows a profound quality in our nature. After pouring out all the venomous feelings and thoughts about this person into this composition, we are encouraged to reread it every few days. Despite the fact that in the heat of the moment we were able to pen such hateful, hurtful words, a few days later those words will seem to be alien and foreign, and we would have a hard time believing that we actually wrote them.
With every subsequent reading, we'll comment "wow, I was being pretty harsh. He's not like that all the time..." and we will begin to see flaws in our hateful view of this person. Moreover, we will begin to find ways to counter the arguments made in the letter, and find redeeming qualities in this fellow as we revisit this letter. Eventually, the feelings of hate and anger will have been replaced by a powerful desire to reconcile with this person, and like the rebbe says, we will run to make peace with him.
This idea enables us to bring the conceptual understanding of mechila into a more concrete form, out of potential into action.
* English adapted from A Student's Obligation by Micha Odenheimer