I think that the previous thoughts are very strong. I would add my own take on Achilles. The first word I would use to describe him is "arete," which is Greek for unparalleled excellence. Usually used to describe honor on the battlefield, Achilles' arete is an essential part of his character. It is what drives him in battle and is the differentiation point between he and all others. I would also use the term "intense" in describing him during his battle with Hector. He is intense on avenging Patroclus' death and displays this throughout his battle with Hector, as he unrelenting in his pursuit and domination over his adversary. Finally, I think that the term "complex" might also be good in describing his character makeup. On one hand, he is incredibly brutal and savage, yet at the same time, he displays compassion to Briseis and to Priam in honoring his first born.
I will go with the following three words: angry, arrogant and loyal.
Achilles is a person who is capable of great anger. In fact, the very first lines of the Iliad refer to his anger. The goddess is asked to sing of this anger and its effects.
He is arrogant in that he thinks that he is bigger than the entire war that the Greeks are having. If Agammenon is going to take Briseis, Achilles is going to be too arrogant to help the Greeks.
Finally, he is loyal -- at least to his friends. He fights and kills and eventually dies to take revenge for Patroclus's death.