In Act 4, Scene 5, Laertes confronts Claudius, seeking revenge for the death of his father. Claudius and Gertrude manage to calm the young man down a little, but then Laertes' mad sister Ophelia returns and reignites his passion. This is one of the places in Hamlet where Claudius displays his intelligence and self-possession. Although he is an usurper, he is such an eloquent and resourceful man that one can understand how he managed to get himself elected king after the death of his brother. One can also understand how he managed to persuade Gertrude to become his wife. She is very supportive of Claudius throughout his critical meeting with Laertes in Scene 5, but, significantly, she is not present during their second meeting in Scene 7 of the same act.
Laertes, I must commune with your grief,
Or you deny me right. Go but apart,
Make choice of whom your wisest friends you will.
And they shall hear and judge 'twixt you and me:
If by direct or by collateral hand
They find us touch'd, we will our kingdom give,
Our crown, our life, and all that we can ours,
To you in satisfaction; but if not,
Be you content to lend your patience to us,
And we shall jointly labour with your soul
To give it due content.
Let this be so;
His means of death, his obscure funeral--
No trophy, sword, nor hatchment o'er his bones,
No noble rite nor formal ostentation--
Cry to be heard, as 'twere from heaven to earth,
That I must call't in question.
So you shall;
And where the offence is let the great axe fall.
I pray you, go with me.
Shakespeare does not actually show his audience how Claudius manages to convince Laertes of his innocence in the death of Laertes' father Polonius. At the end of Act 4, Scene 5 Laertes has calmed down sufficiently to agree to Claudius's proposal to hold a hearing before Laertes' own wisest personal friends and counselors. Claudius should really have no trouble explaining that he is innocent because he is in fact completely innocent. Claudius was appalled when he heard about Hamlet’s stabbing Polonius in Gertrude’s bedchamber while the old man was hiding behind the tapestry. He knew there would be trouble, but he seems fully capable of dealing with all sorts of troubles. He even seems to enjoy it. After all, he wanted to be king, and troubles go with the job.
Then at the beginning of Act 4, Scene 7 it is obvious that Claudius has managed to convince Laertes of his innocence. Not only that, but Claudius now sees how he can use the young man to get his stepson Hamlet killed without appearing to be implicated himself.
And so have I a noble father lost;
A sister driven into desperate terms,
Whose worth, if praises may go back again,
Stood challenger on mount of all the age
For her perfections: but my revenge will come.
Break not your sleeps for that: you must not think
That we are made of stuff so flat and dull
That we can let our beard be shook with danger
And think it pastime.
The stage is set for the final act in which Laertes and Claudius will try to kill Hamlet during a fencing match, while Hamlet himself knows that he finally has to take quick, direction action against the King because news will shortly be arriving that Rosencrantz and Guildenstern have both been beheaded by the English, acting on the documents that Hamlet forged aboard the ship.