Look at the personalities of Achilles and Agamemnon in "The Iliad." What are they like?
I think that the study of both Achilles and Agamemnon represents character sketches of the desire for greatness. Both men seek greatness, and to be recognized for both its pursuit and accomplishment. They are unrelenting in their respective drives and it is for this reason that they collide so often in Homer's epic. Both believe themselves to be inherently superior to their contemporaries and are not afraid to show it. The primary difference between both of them is where they see greatness lies. Agamemnon does not conceal his belief that leaders and political kings represent greatness, and that armies and soldiers follow. Achilles' view is a bit different in that the glory of the nation lies in the exploits of the soldiers; national glory is only possible with the glory of the nation. I think that this difference is what causes the friction between them. Based on the fact that Homer focuses so much on Achilles' transformation and his evolution at the end of the epic, he probably ends up supporting Achilles' viewpoint on greatness over Agamemnon.