To me, there are two major aspects of Achilles' character that stand out in The Iliad. They are his self-centeredness and his obsession with (mostly his own) honor.
Both of these can be seen in the famous story line where he sulks in his tent. To Achilles, his honor is more important than anything else. When Briseis is taken away from him by Agamemnon, Achilles will not fight. It doesn't matter to him that he is the greatest warrior and that it would really help his comrades if he would fight. He will not because he feels his own honor has been insulted. He only comes to fight when his friend Patroklos is killed.
By doing this, he shows that all he really cares about is himself and his honor.