Hamlet feels a great responsibility in that his father has warned him about Claudius' murderous deed. Although this is just an apparition that he has seen, Hamlet realizes that he must fulfill the duty his father has laid upon him. He must avenge his father's death.
As a son, Hamlet is grief stricken. He has lost his father in death, and now he is suspicious of Claudius as his father's murderer. Hamlet feels it his moral duty to avenge his father's death. Also, Hamlet is dealing with the fact that his mother has married her husband's brother. Hamlet feels this is an incestuous relationship. He is abhorred at the whole idea. He has a natural desire to love his mother but is having difficulty with the fact that she has married Claudius only two months after King Hamlet's death. Young Hamlet is torn between duty to his father's memory and the relationship he now has with his mother. This strains the relationship he has with his mother.
Hamlet also carries the role as heir to the throne. As his father's son, he has a responsibility to his country and its people. He realizes that Claudius is corrupt. He knows that something has to be done to rid his country of Claudius. This places a grievous burden on Hamlet.
As for the role of being his mother's son, he is terribly upset with her. He mentions that he is so overwhelmed at her marriage until he desires suicide:HAMLET:
O, that this too too sullied flesh would melt, Thaw and resolve itself into a dew, Or that the Everlasting had not fix'd His canon 'gainst self-slaughter! O God! God!(135) How weary, stale, flat and unprofitable Seem to me all the uses of this world! Fie on't! ah, fie! 'tis an unweeded garden That grows to seed; things rank and gross in nature Possess it merely. That it should come to this!(140) But two months dead!Hamlet is a troubled son with good reason.