In certain respects, it can be awkward trying to justify Achilles's behavior, given that, by modern sensibilities, the actions taken by the Greeks would be classified as atrocities. Remember, as the Iliad opens, we see the Greeks taking women into sexual slavery. When Agamemnon is pressured to release his own preferred captive (whose father was one of Apollo's priests), he decides to take Achilles's captive, Briseis, instead. By modern standards, this behavior is indefensible (with both Achilles and Agamemnon culpable, along with the entire society they represent).
That being said, Achilles does have legitimate grievances to make. Agamemnon, in this scene, shows a prideful and self aggrandizing personality (one could perhaps even call it megalomania), as he uses his own status as supreme commander of the Greeks to try to assert his superiority over his peers.
Ultimately, however, the various Greek heroes and rulers that had entered the war voluntarily, bringing their own followers with them would have been, both from a political and a social perspective, closer to equals than anything else. They are not Agamemnon's subjects, even if he insists on treating them as if they were. Within this context, I don't think Achilles is wrong to view himself as having been mistreated by Agamemnon, given his own status among the Greeks.
Therefore, Achilles complains to his mother, Thetis, asking her to appeal to Zeus on his behalf, requesting that he assist the Trojans against the Greeks, to turn the war against his former allies (and so vindicate himself). Thetis agrees and makes her appeal, with Zeus acceding to her request.