How do Claudius and Polonius view the meeting in Act 3 of Hamlet?
The meeting referred to in this question is of course the meeting that is engineered by Polonius between Hamlet and his daughter, Ophelia, whilst he and Claudius look on, eavesdropping. Through this encounter, they hope to discover the true nature of Hamlet's malady, and whether his madness is as a result of his love for Ophelia or not. Having viewed Hamlet denounce Ophelia and rage at her, it becomes clear to them both that his present madness is not because of his frustrated love for her. Polonius feels that "neglected love" was at the heart of it, but Claudius, having discounted love as the reason for his nephew's madness, now suspects that something much more worrying may be afoot:
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 thus feels that having viewed Hamlet show such rage to Ophelia, he might be plotting some act of violence that could threaten himself. This is when he determined to send Hamlet to England in order to give him a focus to hopefully bring him out from himself. What is key to focus on here, however, is the way that Claudius implicitly recognises Hamlet is a danger to his own position, and therefore he needs to be removed. This is something that is only confirmed by the following scenes with the play that Hamlet organises and then with his slaying of Polonius.