Hamlet Act IV, Scene vii Summary

Inside the palace, Claudius finds out that Hamlet (much to his surprise) is alive and coming home.
Claudius is full of flattery and praise for Laertes, which is a super way to get the young man to be the fall guy for killing Hamlet.
Claudius talks of a noble Norman who spoke respectfully of Laertes, saying he was a fantastic fencer. Claudius claims this really infuriated Hamlet, who wanted to challenge Laertes to a duel.
Claudius hesitates here, asking if Laertes really loved his father and wondering what he might be willing to do to prove it. Laertes announces he'd cut Hamlet's throat in a church, if need be.
Claudius eats it up and hatches a plan: Once Hamlet gets home, Laertes will keep to himself, and everyone else around will be full of praise for his fine sword skills. Claudius figures they can get Hamlet to agree to have a duel. He's even willing to put a little bet down on the fight, which might help to convince the Prince to join in.
Claudius says that because Hamlet is trusting, he's unlikely to really examine the different swords available to the men in the fencing match. Given this oversight, Laertes can choose a sword that isn't blunted (dull swords were used for these friendly duels), and then, tricky like a fox, plunge it into Hamlet.
The guys then go into planning overkill. Just in case Laertes can't kill Hamlet with just a sharp sword, they have a Plan B: a handy-dandy ointment of death obtained from the local mountebank (a traveling quack doctor). If Laertes dips the tip of his sword in the ointment and then stabs Hamlet, Hamlet's sure to die, even from a little nick.
Claudius's final contribution to this scheme is a Plan C: his own drink will be poisoned while he watches the match. As the duel progresses, Hamlet's likely to get hot and thirsty. Then, Claudius will offer up his poisoned drink.
Gertrude enters to inform Laertes that Ophelia has drowned in a brook.
Gertrude's description of the occasion is sadly beautiful: Ophelia went to the brook with garlands of flowers, intending to hang them on the boughs of a far out tree as though they were crowns. A branch broke beneath her, and she tumbled into the brook.
At first, Ophelia's clothes made her float, so she sang old songs and generally appeared like a singing mermaid, without even thinking to cry for help. But her clothes became soaked and pulled her down into the brook, still singing.
Gertrude presents Ophelia's death as a kind of accident, but it may also have been a suicide. It's also not entirely clear how Gertrude knows all of this. Was she there when it happened?
Laertes, hearing of Ophelia's death, calls himself a "woman" for crying over his sister and leaves to be alone.
Claudius comments to Gertrude that he worked very hard to calm Laertes, and he's afraid this will make the kid flare up again.

Act V, Scene i

Act IV, Scene vi