throne, not Claudius.  So, politically, Hamlet is a threat to Claudius. And, he seems to waste no time, upon seeing Hamlet's odd behaviour, in deciding to get to the bottom of Hamlet's actions.  He couches it in a concern for Hamlet's mental health, but his intent seems pretty clear:

. . .I entreat you both
That, being of so young days brought up with him,
. . .That you vouchsafe your rest here in our court
Some little time, so by your companies
To draw him on to pleasures and to gather. . .Whether aught to us unknown afflicts him thusThat, open'd, lies within our remedy.

These final lines of his speech indicate that he wants them, like a pair of spies, to give him information about Hamlet intentions.

Gertrude also supports the idea that it's business and not friendship they are plotting with R & G when she apparently offers them money for any information that bring the royal couple.  She says:

. . .If it will please you
To show us much gentry and good will
As to expend your time with us awhile
For the supply and profit of our hope,
Your visitation shall receive such thanksAs fits a king's remembrance.

Gertrude, like Claudius, is very good at couching this crass request in polite and oblique terms, but the events of the rest of the play (when R & G are exposed as basically the King's henchmen) bear it out.  The King and Queen are engaging the services of R & G as spies in Act II, scene ii.  And the larger significance of this scene is that it is quickly becoming apparent in this play that Hamlet can trust almost no one (except Horatio) at this court.

The conversations between Hamlet and the other three transpire soon after the players have presented their play, 'The Mousetrap' at Hamlet's urging. He had noticed Claudius' reaction to the murder scene which exactly reflected the manner in which his father had been assassinated. Claudius was very upset and rushed out, which to Hamlet, was proof of his guilt. Guildenstern asks to speak to Hamlet to inform him that the king had taken ill and that his mother had sent him to Hamlet, supposedly to discover what was troubling him.

Rosencrantz's purpose is to tell Hamlet that his mother is very upset about the fact that he has insulted the king by staging the play, and that she therefore wishes to talk to him. Hamlet, realising that his two so-called friends have become pawns in the hands of Claudius, plays word games with them, using ambiguity and double entendre to confuse them even more. He toys with them because he realizes that they are being deceitful and are being manipulated by Claudius.

Hamlet distracts Rosencrantz from the main purpose of his enquiry, and starts talking about playing a recorder, which he asks him to do. Rosencranrz declares that he has no knowledge of the instrument and cannot play it. Hamlet's speech hereafter pointedly indicates his frustration and anger with his so-called friends:

Why, look you now, how unworthy a thing you make of
me! You would play upon me; you would seem to know
my stops; you would pluck out the heart of my
mystery; you would sound me from my lowest note to
the top of my compass: and there is much music,
excellent voice, in this little organ; yet cannot
you make it speak. 'Sblood, do you think I am
easier to be played on than a pipe? Call me what
instrument you will, though you can fret me, yet you
cannot play upon me.

Hamlet most clearly expresses that he is insulted by the two men's assumption that they can 'play' him when they cannot even play such a simple instrument. They believe that they can get to the depths of his emotions, yet they cannot even get as much as a sound from the pipe. He says that they may be able to play on his strings, but they cannot 'play' him, i.e. they cannot fool him so easily. He is disgusted by their foolhardy attempts to get to what drives him.

When Polonius arrives, Hamlet plays the same game with him. He makes a fool of the old man who mindlessly and like a typical sycophant, agrees with whatever he says. When Polonius informs him that Queen Gertrude wishes to see him he replies that he will be there soon, it is an easy task. He further states that they (either those he is currently speaking to or Claudius and the queen) will easily agree to whatever he tells them, just as Polonius has just done.

Hamlet now realizes that they are all plotting against him and he also feels that, on this night, evil is afoot. The night breathes out all its evil and he is at a point where he could commit the greatest depravity. He however promises that he would not bring harm to his mother. He might chastise her with his tongue but he would not physically hurt her, like the evil Nero did when he murdered his mother. Hamlet's determination is clear. He has now finally made up his mind to take action. He is not driven only by Claudius' actions, but also by those of his mother, Rosencrantz and Guildenstern, as well as Polonius.