To criticize the protagonist of Sophocles's Oedipus-Rex, we must first examine that character's virtues and vices and comment on how believable and relatable that character is within the story. Let's look at this in more detail.
Our first task is identifying the protagonist. That is the title character himself, King Oedipus of Thebes. He is married to Jocasta and has fathered four children; he is a hero to the people of his city, for he once saved it from the Sphinx. But now Oedipus faces another challenge. There is a plague in Thebes, and an oracle has explained that it is because a murderer has gone unpunished: the man responsible for the death of the previous king.
Oedipus makes a grand show of dramatizing the consequences the murderer will receive. There is something Oedipus does not know, however, and as he moves closer and closer to the truth, Jocasta tries to stop him from learning the reality of the situation. Oedipus himself is the murderer; he fulfilled a prophecy detailing that he would kill his father and marry his mother. By the end of the play, Oedipus blinds himself and is committed to wandering as a lowly beggar as punishment for what he has done.
We can see some of Oedipus' virtues even in this brief summary. He is clever, and he seeks the truth. Yet we can also notice some of Oedipus' vices. He is a proud man, even arrogant at times in the way he treats people. He can be stubborn, and he has a hot temper as we can see in the description of his encounter with the previous king. Had Oedipus simply stepped back and let the party pass, perhaps he would not have fulfilled the prophecy at all. However, he chose to avenge an insult. We could also comment on Oedipus' near despair by the end of the play and his rash act of blinding himself.
As for Oedipus' believability and relatability, we can note that he fits well into the Greek idea of kingship, yet he is also a flawed character. This, perhaps, makes him more relatable. He has plenty of vices, but of course, so do most human beings. Oedipus draws us into his story. We applaud him at some points and become almost angry with him at others, and this is the sign of a well-drawn character. He means something to the audience. We end up caring what happens to him.