The huge gap between being a god and being a mortal is felt pretty keenly in The Bacchae. The god, Dionysus, makes mince meat of all the puny humans who oppose him. Pentheus, Agave, and all the rest of Thebes are all helpless in the face his holy wrath. The god's unchecked unstoppable power is essential in defining his character. The inability of the human characters to do anything to stop him highlights not only their own helplessness, but also the helplessness of all human beings in the face of nature.
The word "drama" comes from the Greek word for action. Given that, it's probably no surprise that most characters in most dramas are defined by their actions. The Bacchae is no exception to this. For example Dionysus's complex character is shown through the things he does. The god is capable of inspiring beautiful harmony with nature as well as horrible violence. King Pentheus's actions also show just what kind of guy he is. He spends the whole play, trying to squelch Dionysus's wild rituals. This action seems to show the King's dedication to law and order, and reveals his stiffly logical mind.
Dialogue is pretty much the only tool a playwright has available to create character. Well, there are stage directions, but ancient Greek playwrights never wrote them down. (Note: If there are stage directions in the version you read they were probably put there by the translator or editor.) In any case, you can definitely get a sense of who the characters in The Bacchae are by the way they speak.
Some good scenes to look to for examples of characterization through speech are the ones between Dionysus and Pentheus. The god speaks mostly in riddles and elusive statements. This not only shows his skill with language, but also highlights his wily nature. Pentheus, on the other hand, is no match for the god's verbal gymnastics. He mostly responds to the god in direct, surface-level questions. This difference in the way the two characters speak directly reflects the contrast between the two characters.