The question of personal responsibility for one's divinely predicted destiny stands at the (complex) heart of this play.
Oedipus acts according to his own nature in the play. He is brilliant, but impetuous. He is a great leader, but he fails to recognize when he should accept the counsel of others.
Stubborn and quick to anger, Oedipus kills his own father on the road to Thebes. This act leads Oedipus to great success. He becomes the ruler of Thebes and saves the city. He marries the queen, Jocasta, and happily sires children. In attaining these things, Oedipus is acting according to his nature.
Although his success is a direct result of "being true to himself," Oedipus' suffering is also a result of the same adherence to the mechanisms and impulses of his personality. These ideas point to the complexity of the question at hand when considering the character of Oedipus, as his strengths are admirable yet they also become his weaknesses.
Ultimately, the answer to the question of how much Oedipus is responsible for his suffering is an open one. There are at least two ways to recommend opposite answers.
First, the play's central commentary may be taken to be one that points to the limits of mortal knowledge (and therefore power to dictate one's own future/life). The gods are in charge, we may say, of Oedipus fate. They made him the person he is and they set his path at birth. Despite his attempts to escape the prophecy, Oedipus fulfills the gods' decree.
Jocasta speaks to this point.
Why should a person fear when the ways of fortune
are supreme, when there is no clear foresight?
This sentiment may be seen as justification for a view that Oedipus is not truly responsible for his suffering.
Secondly, the actions taken by Oedipus are intended to evade or defy a divine prophecy. His hubris and obstinacy lead him to believe that he can dictate the course of his life and avoid a pre-ordained destiny. Coupled with his defiance of prophecy, Oedipus stubbornly insists on having his own way. When others, like Tiresias, tell him to give up his investigation, Oedipus refuses to listen. He instead acts with hubris, again, and suggests that no matter what truth may come he will be capable of handing it.
The depth and boldness of his pride are, in this view, Oedipus' fatal flaw (his hamartia) and in this way all of his suffering can be taken as a product of his willfulness, his inflated sense of power (in opposition to the gods) and in his belief in the ability of an individual to determine the course of his own life.