Because Oedipus the King is a play, Sophocles relies most heavily on action to characterize the players. For example, Oedipus’s fervent pursuit of the truth about Laius’s murder and his own identity reveal both his determination and arrogance.
Dialogue is an extremely important means of characterization here because, as a play, the thoughts and opinions of characters are conveyed largely through speech. For example, Oedipus is portrayed as arrogant because of the tone he uses when speaking to the prophet Teiresias. The way in which characters speak (such as meekly or authoritatively) reflects both their status and confidence.
The qualities of each player are further explored by the commentary of the Chorus. The two halves of the Chorus, the Strophe and Antistrophe, for example, debate the strengths and weaknesses of Oedipus and Creon and by doing so, highlight characteristics of the two men.
Oedipus’s name is pun central. First of all, you probably already know from reading the play that Oedipus means "swollen foot." But if you split the word up differently, you get "know where?" (in Greek). As in, Oedipus doesn’t know where he’s from. Ha! What a gag. Finally, if you decide to chop the word up a third way, you can get (again, in Greek) "alas, two-footed?" The whole foot thing is back (we’re referring to the scars on his feet), but this is also a reference to the Sphinx’s riddle that Oedipus originally solved when he came to Thebes. The Sphinx asked what walks on four legs in the morning, two in the day, and three at night. Oedipus very cleverly answered "a man!" and then there was lots of champagne and further rowdiness. The deal is, a man crawls when he’s an infant and then walks on two legs as an adult and then has a cane when he gets old. So Oedipus’s name, "alas, two-footed," is like saying "Oh no, I’m a man with two swollen feet thus pointing out that I am my wife’s son." It’s also, quite certainly, a tool of characterization.