This is a great question and one that is provocative. Here are a few points that you might want to consider.
First, Achilles and Oedipus are both the main characters of the respective works. They both drive the movement of the literary works. They are both heroes in their own sense and more importantly they are both tragic.
Second, if we focus on the idea of the tragic hero, then we will be able to gain some insights. As with all tragic heroes, both Achilles and Oedipus are extraordinary men. They are leaps and bounds ahead of others. Achilles is the greatest warrior of the Greeks and Oedipus is a very able leader who in the past solved the riddle of the sphinx. The seeking of greatness can be seen as a universal code of morality.
Third, it is their great talents and skills that lead to their downfall. Achilles cannot resist a good fight and the seeking of glory, which ultimately leads to his death and Oedipus must seek to solve all problems, even if he is the cause of it.