Achilles' judgement

In Iliad 1.53/4 Achilles, in response to the bad situation the Achaeans are in, calls an assembly. It is Hera(2) that put him up to it. But he does not call the council first, as is absolutely necessary in the Greek politics of the day. He makes proposals (let's go home or ask a seer) without even talking to the king first. These acts alone would brand him as a would-be tyrant in any polis of the archaic age. With typical Homeric irony, the poet reverses the roles in the scene where Agamemnon wants another prize and Achilles says "when we take Troy" (Il 1.127-9). The normal order is of course the king saying "follow me and you will get your reward when we take Troy". They come for a promise of loot and honor.
Basically, the whole quarrel is because Agamemnon cannot but see Achilles' action as an attempt to be the leader.
So that is his first attempt: power. As he learns quickly, he cannot have it.

We have seen that Achilles cannot have power, Agamemnon already has it. In book 1, the first assembly, when Achilles is on the point of drawing his sword to kill Agamemnon, Athena intervenes. She is sent by Hera who is worried about both men: that makes sense because Achaean power is at stake here: they will never sack Troy if they start fighting among themselves. She convinces Achilles not to draw the sword and promises threefold gifts if he obeys her. His common sense wins out against his emotions and he does what she says. All this sets in motion the Plan of Zeus, where Achilles will stop taking part in the war, the Achaeans will lose because of that until they beg him and offer gifts to come back, thus restoring his honor and teaching Agamemnon a lesson. But, again, what about Achilles?

In book 9, the embassy to Achilles, he appears to get what he wanted: they offer him gifts, they do the begging but still he refuses, leaving his comrades quite stunned. He refuses to fight and threatens to go home to Phthia.

Thetis had prophesied to him that he had a choice of staying at home and gain no honor or going to Troy and have a short but glorious life (Il 9.410-, recalling Euchēnor, boaster, who had a similar choice: go to Troy and be killed or stay at home and waste away from a terrible disease, Il 13.663-. The disease of course is shame.)
A new version of Zeus' plan is born: Achilles will stay out of the fighting, Hector will breach the Achaeans' wall and set fire to the ships, making Achilles send his friend Patroclus instead of going himself. Hector will kill Patroclus and that will finally bring Achilles back into the battle. Achilles' motivation in all this is never explained by Homer (he never explains anything, he just shows the pictures). On the surface it's just continued anger with Agamemnon. But careful reading shows (e.g. 9.401-) that the real reason is simply that he does not want to die. He is going for Athena's offer which implies that he should survive to enjoy his loot (9.364-). This could never be said out loud in a warrior society, nor could Homer write it about their hero but he makes it clear: Achilles would like to be a Diomedes.