He reveals that Claudius murdered him. He doesn't tell Hamlet anything about Gertrude he didn't already know.
No, the ghost instructs Hamlet to seek revenge on his uncle, King Claudius, for murdering him and usurping the throne. The ghost does not mention punishing Gertrude, Hamlet's mother.
His brother, Claudius, came out into the garden while King Hamlet was sleeping and poured poison into his ear.
Hamlet's uncle Claudius killed Hamlet's father (called Hamlet Sr.). The ghost of Hamlet Sr. comes back from the dead and tells Hamlet Jr. so. Hamlet Jr. confirms this by re-enacting the murder as a play, which upsets Uncle Claudius.
The only characters in the play Hamlet who are related to Hamlet are the ghost of his father King Hamlet, his mother Gertrude and his uncle Claudius. None of the rest are related to him.
Hamlet doubts the ghost's statement that Claudius had murdered him--the ghost could be lying. He re-enacts the murder in a play and has it performed in front of Claudius. Sure enough, it causes an attack of conscience in Claudius.
The ghost tells Hamlet the secret of his father's death. Hamlet's father was not killed by a serpent, but was murdered with poison. He also told hamlet that it was his uncle who murdered his father, and that he also seduced Gertrude, his mother. He orders him to revenge his murder against Claudius but to leave Gertrude alone.
In Shakespeare's "Hamlet", The Ghost demands Hamlet kill Claudius, the brother who betrayed, murdered and stole his throne and wife. However, this murderous vengeance does not extend to his offending wife. He tells Hamlet not to seek vengeance on her and to just focus on dealing with the usurper.
Gertrude. I suppose he may have told Hamlet not to take revenge on Gertrude because the ghost still has affection for her, or because it would be particularly evil for a son to kill his mother, or because the ghost knows that Gertrude was not privy to Claudius's murder, and the murder is what he wants revenge for. Or some combination of these.
The ghost tells Hamlet the secret of his father's death. Hamlet's father was not killed by a serpent, but was murdered with poison. He also told hamlet that it was his uncle who murdered his father, and that he also seduced Gertrude, his mother. The ghost then tells Hamlet to avenge his father's death but to leave his mother alone.
Hamlet's father is dead. His uncle (Claudius) marries his mother. Hamlet sees his father's ghost and the ghost tells him that Claudius murdered him. Hamlet seeks revenge and kills Claudius. Hamlet also dies. There are many others plots in Hamlet but that is the main one
Hamlet is disposed to believe ill of Claudius. He dislikes him and resents his marriage to his mother. That is why when the Ghost names Claudius as his murderer, Hamlet says "O my prophetic soul!"--he had already suspected as much. Because we hear Claudius admit his guilt (just before the "To Be or Not to Be" speech and in Claudius's "O my offence is rank" soliloquy) we know that the ghost is telling the truth. Otherwise, it might be plausible to believe that Claudius is innocent, as everyone including Gertrude seems to believe.
Hamlet is talking to the Ghost, which Gertrude can't see, which looks bizarre to her. For the first time, Gertrude sees Hamlet as seriously mentally ill, not just disturbed. Ironically, it's Gertrude whose perceptions are faulty in this scene.