What conclusion did the King come to regarding Hamlet after hearing Hamlet and Ophelia's conversation?Act III; scene i
To establish the context for your question, it is important to consider what just happened. Hamlet, suspecting that he is being watched and perhaps even being "set-up" by Ophelia, has commanded "get thee to a nunnery!" He explicitly states that he wishes she would remove herself from the sinful nature of men and "not be a breeder of sinners." Later in the conversation he implies that she is untrue, saying, "God hath given you one face, and you make yourselves another." He also claims to have never loved her and never given her any gifts of his affection. None of this speech is nice -- it is actually quite cruel in places.
Claudius and Polonius have been hiding behind the curtains in the room and have overheard the entire conversation. Polonius has previously tried to say that Hamlet is acting crazy because Ophelia has rejected Hamlet and he is heart broken. Claudius's first words after overhearing their conversation make it very clear that he has his doubts on THAT theory. He says, "Love? his affections do not that way tend." There is nothing loving or longing his Hamlet's words. He goes on to say that "There's something in his soul / O'er which his melancholy sits on brood." He is saying that there is something else eating at Hamlet and causing this unsettling behavior, and he plans to send Hamlet away for awhile, to England, in order for things to settle down a bit for everyone. Claudius is clearly worried in his not knowing the cause of Hamlet's madness because he ends this conversation with the rather ominous comment: "Madness in great ones must no unwatched go." He is on heightened alert now, making Hamlet's cause all the more potentially challenging.