Describe Achilles' character in three words.

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.