What do you notice about Achilles character ?and his philosophy in life
I think that I notice the dualistic conception of intense anger and striving for arete. Homer's characterization of Achilles allows the reader to see someone capable of the highest of glory along with the most intense expression of wrath. Both are on display when Achilles battles Hektor. On one level, there can be a certain level of marvel at Achilles' prowess in the battle with his formidable adversary. Yet, as he is fighting, the undercurrent of anger felt towards Patrocles' death is present. It is almost as if the intensity of his emotion is channeled into a focused exercise of greatness. Such a depiction is rare because the obvious depiction of anger and wrath is that it is all encompassing of the character, leaving no room for anything in way of a focused and discipline demeanor. However, in Achilles depiction, there are moments where one can see both.
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.