Discuss the duel in detail.

Laertes and Claudius have initiated a plan that begins with Laertes' challenge to Hamlet, and Laertes plans to tip his rapier with poison so that if he even scratches Hamlet, the prince will die.  To make extra sure that Hamlet will perish during this duel, Claudius will poison a cup of wine so that, when Hamlet stops to drink, he will be poisoned in this way as well. 

Before the duel even begins, Hamlet sincerely...

Laertes and Claudius have initiated a plan that begins with Laertes' challenge to Hamlet, and Laertes plans to tip his rapier with poison so that if he even scratches Hamlet, the prince will die.  To make extra sure that Hamlet will perish during this duel, Claudius will poison a cup of wine so that, when Hamlet stops to drink, he will be poisoned in this way as well. 

Before the duel even begins, Hamlet sincerely apologizes to Laertes.  After the duel starts, Hamlet gets the first hit against Laertes.  Claudius offers Hamlet the poisoned wine, but Hamlet doesn't want to stop for a drink yet.  He gets another hit on Laertes.  The Queen, his mother, offers him her handkerchief to wipe his sweat, and she picks up the poisoned wine to drink to Hamlet's health; Claudius tells her not to drink, but she does anyway.  Hamlet thinks that Laertes isn't trying his hardest, and so he tells him to fight as hard as he can.  In the next moment, Laertes strikes Hamlet and, in the scuffle, the two switch weapons (and Hamlet ends up with the poisoned rapier) and Hamlet then wounds Laertes.  Gertrude collapses from her poison, and now Laertes says that he feels faint and that he's been caught in his own trap.  Gertrude realizes that she's been poisoned by the drink.  When Laertes explains that the weapon's been poisoned too, Hamlet stabs Claudius with it and forces him to drink from the poisoned cup as well.  Claudius dies quickly and Laertes says that the king got what he deserved because he's the one who mixed the poison; Laertes also exonerates Hamlet from the blame for his own and his father's deaths, and he says that he should not be held responsible for Hamlet's death either.  After Laertes dies, Hamlet tells Horatio that he must live to tell the real story after Hamlet is gone.  Hamlet worries that nobody besides Horatio knows the truth, and so Horatio must live long enough to share it.