In scene I, Marcellus notes uneasily the appearances of the ghost. He and Bernardo have brought Horatio out at night to talk to it, thinking he may, as a scholar, know how to approach it. Marcellus also notes that the appearance of the ghost coincides with a build-up in the shipyards: the ship builders, he says, are working seven days a week, and "brazen cannon" is also being "daily cast." He asks Horatio what that is all about. Horatio explains the back story of the war between the senior Hamlet and Fortinbras, and notes that the young Fortinbras, to avenge his father's death, is planning to march on Denmark. Denmark is preparing for war. Bernardo ties this as well to the appearance of the ghost and sees the ghost as a bad omen: he reminds his listeners of similar signs before Julius Caesar was murdered, saying "the sheeted dead / Did squeak and gibber in the Roman streets."

In scene II, we learn of Hamlet's deep grief and troubled relationship with Claudius. Hamlet calls Claudius "a little more than kin and less than kind." At the end of the scene, Horatio, Marcellus and Bernardo arrive to talk privately with Hamlet about the ghost of his father. 

From Marcellus's point of view, the chief sign of something being awry in Denmark is the repeated and frightening appearance of the ghost, which is why he wants someone of a higher class than himself to address it. When it will not talk to Horatio, he and the others appeal to Hamlet, who says he will speak with it "though hell itself should gape / And bid me hold my peace."

What is rotten in the state of Denmark is regicide, the killing of a king. It was believed in Elizabethan times that this was one of the worst possible types of murder - it upset the very fabric of existence. Plus, in this case, not only has the king been murdered, but his brother is the murderer and that very brother has now married the dead king's widow!

In Scene I, Horatio, Marcellus and Bernardo are standing watch and they see the ghost of Hamlet's father, the dead king that has been killed by his brother, Claudius, who is now king. In Scene II, Hamlet is talking to his mother, Gertrude, and his uncle, Claudius, who is now king. Hamlet's father is dead and within a very short period of time, Claudius has married Hamlet's mother. Hamlet is very depressed by this, thinking that his mother has not honored his father's memory by marrying so soon. Gertrude and Claudius are worried about Hamlet's mental health. Later, Horatio, Marcellus and Bernardo come in and tell Hamlet that they have seen the ghost, so Hamlet resolves to watch for the ghost himself that night.

You can read these scenes right here on eNotes.