It will be difficult to describe Achilles with only two adjectives, as he is a complex character. However, if I had to choose two adjectives, I would say that Achilles is proud and wrathful.
Achilles is proud in the sense that he knows his honor and when it is slighted by Agamemnon, Achilles decides he will not fight. Even when the Greeks are dying, he will not fight, because his honor has been bruised. Even when Agamemnon offers his vast amount of riches, Achilles will not yield. The reason for this is that Agamemnon would still be higher. Achilles's pride will not allow for this.
Second, Achilles is wrathful. We see this in his treatment of Agamemnon, but we also see this even more in his dealings with Hector and the Trojans. When Hector kills Patroclus, Achilles's wrath grows exponentially. At one point, he won't even give Hector a proper burial.