This is a question that the play itself asks. Is Oedipus dominated by fate and therefore powerless to avoid doing exactly what he does or is he able to make choices for himself and therefore responsible for his actions?
One thing that is clear in the play is that Oedipus puts forth a great effort to escape the fate that is prophesied for him. He is told that he will kill his father and sleep with his mother so he flees from his home. A question arises here as to whether or not Oedipus was exhibiting pride in this effort. Is it hubris for a man to believe that he can escape a doom that the gods have decided for him? If we take the view that it is pride driving Oedipus to escape his fate, then we can say the ultimate end of his story in the play is deserved and is derived from this character flaw.
However, if we presume that it is only natural to try to escape such a terrible fate then we have to look further into the play to see if Oedipus is responsible for what happens to him.
Oedipus' greatest strength is his strength of will. This is what enables him to solve the riddle of the Sphinx (insofar as will and the mind are linked) and it is also what makes him a natural ruler. As king, Oedipus imposes his will on others. In fact, this is how he became king. We might find some fault in the way Oedipus chooses to kill the old man he meets on the road to Thebes, running him off the road and imposing his will instead of allowing the older man (his father, the king of Thebes) to pass.
If Oedipus were a man of different character, he may have refrained from killing the man on the road. We might reasonably argue that if a person like Oedipus were really out to avoid patricide he would go to great lengths to avoid killing anyone, especially anyone of an older generation. Here again we might find a character flaw that can be attributed to Oedipus and thus make him a candidate for ultimate responsibility for his end.
Is this Karma? We might say that in his pride and his uncompromising disposition Oedipus has committed negative acts that will, in the way of Karma, lead to negative ends. The choices to flout the dictate of the gods and to kill a stranger on the road each contribute to the fulfillment of the prophesy.
The question remains, despite these prevarications, should we blame Oedipus for his actions or should we empathize with his plight? He is acting on a rather pure mission - to avoid a terrible fate. We might ask then if the choice of the gods to place this fate before him is the "first cause" in the Oedipus story that functions as the initiating influence on his future.
"In time you will know this well:
For time, and time alone, will show the just man..."