Does Hamlet succeed in fulfilling the three tasks assigned to him by the ghost (his father)?
In an answer, Yes. There are some that may say that Hamlet did not honor his father's ghost wish to "nor let thy soul contrive against thy mother aught". But, overall, despite his intense anger towards his mother for marrying Claudius, he did not seek advertant revenge on her. He did however, avenge his father's death by killing Claudius, remember him in love and loyalty, and did not truly mean to punish Gertrude for marrying Claudius. It just happened that she drank from the poisoned cup that was meant for Claudius only.