There are two main themes that need to be reckoned with by the characters before they are able to heal. Firstly, they must work together for once, and figure out how to let everyone get what they want, instead of reviling their enemies and disrespecting them. Secondly, they need to learn to grieve, and in the context of war, that isn't possible, so there needs to be peace, at least until everyone can get themselves all put together again.
The first theme is Priam's central epiphany. When Iris teaches him about chance, he learns that his pain and misfortune don't necessarily mean that he has been abandoned or rejected by the gods. Knowing that, he has the valor to face the entire enemy army alone, as a humble man. Then there is the messianic image of a king going into enemy territory with a donkey instead of a horse. Priam finds peace on this path, the simple path where a commoner gets to talk his ear off instead of letting him think about his son. He knows before he even arrives that the universe itself has blessed his attitude, because the gods are coming to assist him. Achilles mistakes him for his father, a testament to Priam's simple appearance, and that sets the bedrock for Achilles's willingness to reason with Priam and make a deal.
The second theme is clearly seen on either side of the story, Greek or Trojan, but it's important to note Achilles's story in particular because it makes something really obvious which Priam learned in a quieter, easier way. The path to healing is never the path of vengeance and hatred. When Achilles's allows himself to become enraged against Hector's body, all he accomplishes is the scorn of the gods who have already forbidden Hector's body from taking damage. In other words, the idea that you can beat a dead body and feel better is false—there must be a higher, more ethically virtuous path to healing.
Achilles finally learns what the path to health really looks like when he finally sees the body of Hector for what it is: A tragic loss of life. It isn't Hector that Achilles is really mad at anymore. Rather, his inability to grieve his friend has been the product of his unwillingness to move through his emotions instead of getting hung up on petty things like pride and vengeance. When he learns that Hector's body is meaningful, and that the loss of his life is sad, he realizes that he can have compassion for his enemies, and suddenly, a great weight is lifted from him. What has been afflicting him was hate. Without hate, he can see the world for the tragic, ironic place that it is, and he feels better.