Troy (I do not remember Homer's the Iliad very well, so I'll base this off the movie instead)
Paris: ESFP, all his ESFP weaknesses are glaring
Hector: INFJ, rational, but defintely a Feeler. he's pretty much the only character who actually has any kind of altruistic intentions
Achilles: ISFP, extremely contradictory. on one hand, he is a merciless killer, on the other, he has a sweet romantic side and is a rather independent thinker when it comes to his values. his actions to me scream Fi with the SP fascination of glory, status and impulsiveness secondary to this. he is a good represention of Introverted Feeling tho. he's romantic and has strong values, but he only cares about a select few people. if you go against his values or hurt one of those few people tho, he will unleash the might of the gods and show no mercy to his enemies as witnessed when Hector kills Patroclus.
Priam: dillusional ESTJ or ISTJ. he has moments of wisdom, but overall he has this warped sense of the way things are supposed to be and has little knowledge of how they actually are. he also seems to care more about the gods and the rules of the land than his own sons.
Patroclus: definitely F, but that's all I can say for sure. at times he seems SFJ, at other times INFP or ISFP and still other times ENFJ, which I think is the most likely
Ajax: ENTJ, the stereotypical buff bearded warrior who crushes everyone with his sheer man power =P
Menalaous: ESTJ, sorta like Agamemnon, but without the ENTJ ambition and placing higher value in tradition and "honor" (in other words, Pride)
Agamemnon: ENTJ, although he looses most of his rational powers because of his pride and arrogance which cloud his judgement
Odysseus: ENTP, the only actually rational person in the whole movie who doesn't let his pride get him killed. he also has all the inventive powers and trouble shooting abilities that are trademarks of ENTPs.
Eudorus: ESFJ? definitely SJ, but an extremely bad ass one who doesn't take the rules too seriously. he seems E because he's a wild and crazy killer and F because SFJs tend to follow people as opposed to STJs who follow sets of rules and regulations
Briseis: ENFJ, certainly an Idealist, definitely stark contrast between her extraverted feeling and Achilles introverted feeling
Helen: no clue. If I had to guess, ISFP, INFP or INFJ. definitely not SJ tho