Oedipus was a mythical Greek king of Thebes, a tragic hero in the Greek mythology. Oedipus accidentally fulfilled the prophecy and despite his efforts not to, he end up killing his father and marrying his mother, and brought disaster to his city and family. It is well known that “Oedipus represents two enduring themes of Greek myth and drama: the flawed nature of humanity and an individual's role in the course of destiny in a harsh universe”.

Is Oedipus character his destiny? Is what happens to Oedipus his fault, why or why not?

First of all, yes, Oedipus character is his destiny. Indeed, before the play, the final outcome of the play is known before Oedipus is even born. And in fact, "his life is clouded with the presage of disaster; he was destined to one day kill his father and to become his own mother's husband".
We can say that Oedipus destiny was pre-determined. No matter how good person he was, in one way or another he would have fulfilled the prophecy and end up in disaster. So, due to the prophecy, Oedipus had only a small part in his own downfall.

But, what happened to Oedipus is also his fault. Indeed, he is too determined to find out who he is and it shouldn't be so important to him. He is also too proud to listen to the gods; he thinks he can get out of following through on his fate.
Oedipus was a very proud man. I think his pride was his biggest character flaw and because of this pride, the conclusion of the play was tragic.

To conclude, the King Tiresias told the truth when he says to Oedipus, "you are your own worst