Polonius has assured Claudius that he has diagnosed Hamlet's apparent melancholy and madness. The old man believes that Hamlet is madly in love with his daughter Ophelia, and he persuades the King to spy on both of them when he arranges for them to meet together. The meeting between Hamlet and Ophelia does not evolve as Polonius expected. Hamlet behaves erratically and repulses the poor, innocent girl, telling her, among other things, to get to a nunnery. It would appear that Hamlet feels fairly certain that he is being spied upon by the King or by one or more of the King's agents. It also would appear that Hamlet thinks Ophelia is cooperating with Claudius in prying into his secret thoughts.
After Claudius and Polonius re-enter the scene, the King expresses his dismissal of the love hypothesis and his suspicions that Hamlet is really plotting against him. Claudius uses one of Shakespeare's most beautiful and most characteristically homely metaphors to convey his feelings.
Love! his affections do not that way tend;
Nor what he spake, though it lack'd form a little,
Was not like madness. 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
Claudius is an ambitious, secretive, cunning man, and he naturally judges others by himself. The metaphor Shakespeare puts in his mouth shows that he sees his stepson's melancholy as a big, dark bird, something like a condor, sitting on a single egg which will disclose something frightening when the bird finally spreads its wings.
Polonius is chagrined, not only because the meeting between Hamlet and Ophelia did not come off as he expected, but because he would dearly love to have his daughter marry the heir apparent to the Danish throne. Polonius' perceptions are distorted by his own ambitions. If Hamlet became king, then his daughter would be queen and he himself would be one of the most powerful and socially prominent men in Denmark. Laertes too would benefit, because Hamlet could give him a title and a fiefdom. Polonius responds to Claudius' assessment of Hamlet's feelings about Ophelia and his intention of sending Hamlet on a mission to England as follows:
It shall do well: but yet do I believe
The origin and commencement of his grief
Sprung from neglected love.
Polonius may be a wise old man, but he does not realize who he is talking to. It would never occur to him that Claudius had become king by murdering his own brother. Claudius has the darkest possible thoughts, fears and suspicions because he is riddled with guilt. He knows that if Hamlet decided to assassinate him or to instigate a coup against him, it would only be what he richly deserves. As Macbeth says to himself when he is planning to murder Duncan:
But in these cases
We still have judgment here; that we but teach
Bloody instructions, which, being taught, return
To plague the inventor: this even-handed justice
Commends the ingredients of our poison'd chalice
To our own lips.
At this point in the play, Claudius probably does not plan to have Hamlet executed when he gets to England. It is not until after Polonius is killed in Queen Gertrude's bedroom that Claudius decides to use the mission to England as a means to get rid of Hamlet permanently.
Do it, England;
For like the hectic in my blood he rages,
And thou must cure me: till I know 'tis done,
Howe'er my haps, my joys were ne'er begun. (IV.3)