What does grief and how each handles it say about the characters in Oedipus Rex?
Well, we can certainly make the following observations about Oedipus and Jocasta based on how they respond to the revelation of the true identity of the murder of Laius in the play. The act of Oedipus in gouging out his own eyes is of course tremendously symbolic. He as a character has never truly been able to "see" as he has been blinded by various traits, including his own unwillingness to accept the truth, no matter how many hints and clues were given to him. Now that Oedipus has "seen" the truth of who he is, he is unable to carry on "seeing," or rips out his eyes to try and somehow ignore the truth. This measure of course does not work. The way Oedipus reacts shows how the grief of the revelation causes him to do extreme things without thinking about the consequences.
For Jocasta, we can see similar aspects in the way that she responds to the tragedy. Her act of suicide of course also represents a similar feeling of shock and terror. However, we can analyse this a bit further. Jocasta can literally carry on living with the truth that she married her son and her husband's murderer. This is why she spends so much time trying to get Oedipus to forget his search and to ignore the various prophecies. She cannot live and accept the truth, and thus the grief she feels when the truth is made known causes her to take, from her point of view, the only option she has: to kill herself and thus escape the truth.
There is a sense then in which the responses of these characters seem in some way to prevent them having to face and accept the reality of the grief they face.