What scheme does claudius and polonius have while Hamlet is visiting his mother in her quarters?

Answer 1

It's mostly Polonius's scheme, not Claudius's. Polonius is a spymaster and his answer to everything is to spy on the person. He proposes that by spying on Hamlet's supposedly private conversation with his mother, Polonius may learn something relating to Hamlet's madness. It's an Elizabethan wiretap.

Polonius suggests the scheme in 3,1 as follows:

My lord, do as you please,

But if you hold it fit, after the play,

Let his queen mother all alone entreat him

To show his grief. Let her be round with him,

And I'll be placed, so please you, in the ear

Of all their confidence.

Claudius agrees because "madness in great ones should not unwatched go." In 3,3 Polonius tells Claudius "he's going to his mother's closet. Behind the arras I'll convey myself to hear the process." he then says

And, as you said, and wisely was it said,

'Tis meet that some more audience than a mother,

Since nature makes them partial, should o'erhear

The speech of vantage.

Claudius said no such thing. Polonius is putting his own words in Claudius's mouth and then congratulating the king on how clever he is for saying them. Polonius knows how to be a courtier.