Hamlet seeks justice as he tries to discover whether Claudius is in fact guilty of the crime the ghost has accused it of. He seeks to take the throne away from a murderer, enacting justice in the face of wrongdoing. His journey of revenge is a quest for justice to his father, and potentially his mother if she has been victimized in any way. Justice is also seen as Fortinbras, in the end, is able to take the kingdom from the family who killed his own father.
Judgment is seen in Hamlet's attitude towards his mother for marrying so soon after his father's death. His judgment towards her is not discriminating-it is towards her and ALL women. They are all fickle and unfaithful. He expresses this harsh judgment to Ophelia and later to his mother herself in her bedroom after the play. He judges his uncle for his evil deeds done, Polonius for being a foolish old man, and his friends for conniving against him.
Justification leads us to ask many questions. Is Laertes justified in plotting to kill Hamlet, considering all Hamlet had done to his family? Is Gertrude justified in her marrying the king so fast-do we have the entire picture there? Is Hamlet justified in a number of things: procrastination in enacting revenge, killing Rosencrantz and Guildenstern, his rant against Ophelia, or his final actions?
These three themes are prevalent throughout the entire play and can be discussed in great lengths each. I hope this gives you some ideas!