This is an interesting question, but a difficult one, because of Achilles' character. He is so emotional, so driven by passions that it is hard to imagine him having a philosophy in the sense of a rationally worked out model of the world. That said, in Book 24 Achilles seems to have the following image of or attitude toward the world. First, individual passions trump social norms. He is so grief-stricken over his friend's death that he lets that drive him. He also desecrates Hector's body in ways that warriors of the time would not. (Look at how others are restrained by honor.) Second and implied in the above, pain trumps honor.
Third, familial bounds are more important than those of the larger society. You see this when he gives in to Hector's father's request.