From the very beginning of the play Claudius shows that he is suspicious of Hamlet. Since Claudius usurped the throne by killing Hamlet’s father, it is natural for him to be afraid that Hamlet might be plotting to seize the throne by killing him. Much of the play is about how Claudius seeks to pry into Hamlet’s mind, attempting to find out what he is thinking and feeling. Claudius believes that Hamlet’s mourning and melancholy have a deeper significance than grief for his father. In a beautiful metaphor, Claudius tells Polonius
There’s something in his soul,
O’er which his melancholy sits on brood,
And I do doubt the hatch and the disclose
Will be some danger (3.1)
Claudius tries to use Polonius, Ophelia, Gertrude, Rosencrantz, Guildenstern, and probably others to spy on Hamlet and report back to him. Polonius gets himself killed by spying on Hamlet. Rosencrantz and Guildenstern arouse Hamlet’s suspicions and dislike by trying to pry into his thoughts, and this leads to their deaths when he decides to spy on them. Hamlet pretends to be insane mainly in order to make it difficult for Claudius to understand what is going on inside his mind.
In his long speech in the first act of the play, Claudius tells Hamlet:
For your intent
In going back to school in Wittenberg,
It is most retrograde to our desire,
And we beseech you: bend you to remain
Here, in the cheer and comfort of our eye,
Our chiefest courtier, cousin, and our son. (1.2)
What he means is that he wants to keep Hamlet in Elsinore where he can not only keep an eye on him but can recruit others to do the same thing. If Hamlet were allowed to leave the country, he could get help from foreigners in starting a revolution or even in raising an army to invade Denmark, as Malcolm did in Shakespeare’s Macbeth. Even if Hamlet had no such intentions, he could meet people who would incite him to rebel against the man who stole the crown which was rightfully his. Claudius is already having trouble with Fortinbras, the representative of one foreign power; he doesn’t need any more trouble with foreigners now.
Hamlet has no subversive intentions at the time of Claudius’s speech in Act 1, Scene 3, but he becomes extremely dangerous after he has the conversation with his father’s ghost in Act 1, Scene 5.