What is Claudius and Laertes' three part plan to kill Hamlet?
OK. It's the fencing match. And I think it's actually a two-point plan. First of all, Laertes once to cut Hamlet's throat in the church, but Claudius persuades him there's a better way. To somehow rig the fencing match so that Hamlet ends up dead. Laertes agrees:
I will do't!
And for that purpose I'll anoint my sword.
I bought an unction of a mountebank,
So mortal that but dip a knife in it,
Where it draws blood no cataplasm so rare,
Collected from all simples that have virtue
Under the moon, can save the thing from death
This is but scratch'd withal. I'll touch my point
With this contagion, that, if I gall him slightly,
It may be death.
So, the poison on the rapier will kill Hamlet. But Claudius organises a back up plan:
Therefore this project
Should have a back or second, that might hold
If this did blast in proof. Soft! let me see.
We'll make a solemn wager on your cunnings—
When in your motion you are hot and dry—
As make your bouts more violent to that end—
And that he calls for drink, I'll have prepared him
A chalice for the nonce; whereon but sipping,
If he by chance escape your venom'd stuck,
Our purpose may hold there.
Claudius will put poison into a chalice, and offer it to Hamlet as a toast, if the rapier plan fails to work. So it's a two-point plan for killing him off.
Hope it helps!