The next morning, King Claudius, the brother of the dead king, holds court. He uses pretty language to make his recent marriage to Gertrude, his brother's widow, sound perfectly normal. He says it is possible to balance "woe" and "joy."
Claudius uses language as a tool to smooth over actions that are immoral. He uses language to create the appearance of propriety.
Claudius then says he has received a message from Fortinbras demanding Denmark give up the lands Old Hamlet won from Old Fortinbras. He sends Cornelius and Voltemand with a message to Fortinbras' elderly uncle, the King of Norway.
Fortinbras is a son looking to avenge his father. He takes action, however seemingly foolish, to achieve his ends.
Claudius turns to Laertes, the son of the Lord Chamberlain, Polonius. Laertes asks to be allowed to return to his studies in France. Claudius agrees.
Laertes/Polonius is another father/son relationship.
Next, Claudius turns to Hamlet, and asks why he is still dressed in mourning clothes. Gertrude wonders why he "seems" so upset. Hamlet says he "is" upset, and that his clothes can't capture his true mourning.
By emphasizing that how he "is" is more important than how he "seems," Hamlet implies that his interior reality is more powerful than any appearance.
Claudius chides that it's natural for fathers to die and for sons to mourn, but that mourning for too long is unnatural and unmanly. He asks Hamlet to see him as a father, since Hamlet is first in line to the throne. He asks Hamlet not to return to Wittenberg, Germany to study.
Claudius lectures Hamlet on what's natural, but Claudius murdered his own brother! Appearance vs. reality. Also, Wittenberg was where the Reformation, a schism in religion, started.
Gertrude seconds the request. Hamlet promises to obey his mother.
But, tellingly, he doesn't promise to obey Claudius.
All exit but Hamlet. In a soliloquy, Hamlet wishes he could die and that God had not made suicide a sin. He condemns the marriage between his mother and uncle. He says Claudius is far inferior to Old Hamlet, and, in anguish, describes Gertrude as a lustful beast.
It's important to note that Hamlet's death wish exists even before he learns of his father's murder. Fury at his mother's marriage to Claudius is enough to make him contemplate suicide.
Horatio, Marcellus, and Barnardo enter. Hamlet, who studied with Horatio at Wittenberg, is happy to see his friend, and pleased when Horatio agrees that Gertrude and Claudius's marriage was hasty.
Horatio proves he is willing to speak honestly about reality by noting the speed of the wedding.
Horatio tells Hamlet about the ghost. Hamlet, troubled, decides to watch with the men that night.
Hamlet learns his internal feelings of unease are mirrored by spiritual unease in Denmark.