Tools of Characterization
The most common way in which Homeric characters reveal themselves is through their actions. This is appropriate to the world the Iliad depicts, in which characters regularly accuse each other of talking the talk but not walking the walk (see Achilleus's criticisms of Agamemnon in Book 1).
In this poem, it's mostly what you do, not what you say, that counts. For example, when Hektor boasts in Book 18 that he will never run away from Achilleus, it shows us something about his character, but not what he wants it to show.
That's because, by the time we finish Book 22, we know that Hektor does run away from Achilleus, until he gets his courage back and fights. If his boast in Book 18 shows us that Hektor's pride sometimes gets the better of him, his other actions—such as comforting his wife and child—bring out his complex and sympathetic character.
What about the picture of Achilleus that emerges from his actions? How can the same guy drag a corpse from the back of a chariot and then have dinner with the dead man's father? Homer's nuanced portrayal of actions forces us to ask such questions.
One of the standard ways in which Homer reveals something about his characters is through "epithets." These are small, descriptive phrases that typically get attached to a single character—and then repeated, many, many times.
Some of these are pretty standard-issue, like "lord of men" for Agamemnon and "swift-footed" for Achilleus. Others are more surprising and interesting, like "of the counsels" to refer to Zeus. This epithet reminds us that nearly everything that happens in the story follows the god's sneaky plans.
Not all of the names in the Iliad have meanings attached to them, but some do, and these are important. Some scholars (notably Gregory Nagy, whose recitations of Homer may be found in the "Audio" section of our "Best of the Web") believe that the name Achilleus originally means "pain for the people."
If so, this would be fitting in two ways. It's easy to imagine Peleus, hoping his son would grow up to be a mighty warrior, giving him such a name—expecting that he would be causing pain to other people than his own. This highlights the irony that some of the greatest pain Achilleus causes is to his fellow Achaians. On the other side, Hektor's name derives from the Greek word meaning "to hold." This is fitting because Hektor can be thought of as the "holder-in-place" or "protector" of Troy.
Speech and Dialogue
Characters in the Iliad often reveal something about themselves through how they talk. For example, you know whenever Nestor hits the scene that you're about to hear an incredibly long-winded story about how awesome he was before he became old. Similarly, Achilleus's speeches are usually way over the top, either overflowing with insults, or wallowing in self-pity, or saying things so savage you can barely believe he's human.