Horatio is found hanging around the Danish court after Hamlet leaves. He is present for Ophelia's mad scene and has a short line. He is basically kicking his heels waiting for Hamlet to get back. There is nothing in the play which would suggest that anyone has given him a task during this period.
That is the correct spelling of the male given name "Horatio."(Shakespearean character in Hamlet, or Captain Horatio Hornblower)
Claudius arranged for Rosencrantz And Guildenstern to take Hamlet to England with a letter that says to kill Hamlet. The letter is to be given to the King of England, but Hamlet steals the letter on the boat ride over, and replaces it with one that says to kill the bearer of this letter.
While on board a ship for England, Hamlet took a sneak peek at the sealed orders given to Rosencrantz and Guildenstern.
The incident in which Hamlet gets off the ship to England plays an important role in changing the entire plot. As he was being "escorted" to England by Guildenstern and Rosencrantz, he switches the letter that the King had given with the one he writes. The original message that the King had wanted to pass on contained an order to kill Hamlet in England. However, Hamlet switches the letter with a new one which orders for Rosencrantz and Guildenstern to be executed. After this, a pirate ship attacks the ship that Hamlet was on. We are not given much detail on such an occurrence but we find out that Hamlet got onto the pirate's ship as the pirates were attacking the ship that was supposed to take Hamlet to England. Shakespeare does this in order to bring Hamlet back to Denmark and resume the plot. Although, the literary merit behind such a plot manipulation is widely debated as being either acceptable or flagrantly unacceptable.
By Act 5, Hamlet knows just about everything he needs to know. He reports to Horatio that he discovered the secret orders given to Rosencrantz and Guildenstern which proved that the King was trying to kill Hamlet. He does not yet know about Laertes' complicity in this, but doesn't try to find out either. When he finds out that Ophelia is dead, he does not inquire after the details of her death. He has made a decision to go with the flow ("There's a special providence in the fall of a sparrow.")
Polonius dies. Ophelia dies. Claudius dies. Hamlet dies. Hamlet returns home for his father's funeral. Hamlet stages a play to prove Claudius's guilt. Ophelia drowns in a river. Hamlet and Laertes duel. Hamlet returns to Denmark to bury his father. Hamlet kills Polonius. Ophelia is found dead. Laertes and Hamlet duel.
The official story is that King Hamlet died from snakebite.
Rosencrantz and Guildenstern - both who end up dead.
The mouse trap is the play inside the play. Hamlet crates a play simulating his father's death in front of his uncle. Hamlet expects his uncle to react in order to show that he's the murderer.
He was sleeping in his orchard when Hamlet's uncle, Claudius, poisoned him with hebenon.
The speech that Hamlet gives to the players means don't over act. The speech is given in Act 3 Scene 2 of the play.
Hamlet thinks that Rosencrantz and Guildenstern are foolish little pawns because they do anything the King tells them to because they just want to be honored. Furthermore, he thinks they are just plain foolish because he tricks them and manipulates them multiple times to perfectly stage his plan.