Explain the differences between how the characters of Homer (Iliad) dealt with their monumental struggles as compared to how the characters in Oedipus the King and Euripides Medea.

As there are numerous different characters in all of these works, a comparison of all of them to each other would run to hundreds of pages.
The conflict between the Trojans and Greeks is handled by violence. Paris seized or abducted Helen. Menelaus, the jilted husband, has enlisted the support of other Greeks to reclaim her by violence.
Within the Greek forces, conflicts are normally handled by debate, but Agamemnon also has the power to make executive decisions. Achilles handles his conflict with Agamemnon by withdrawing from the fight, hoping that subsequent losses in battle will compel Agamemnon to return his war prize. Odysseus tends to favor subterfuge.
Oedipus exists within a more rational and legalistic framework. He consults oracles and follows laws. Medea takes justice into her own hands and resolves conflict with violence.