Questions...? Hamlet 3:3

My first question is Does King Claudius think that Hamlet is on to him? Cause as soon as the play ends he wants Hamlet taken away to England. The second question is why didn't Hamlet just kill his uncle? He was right there, he has putting on this madness act so can figure out if the ghost is lieing and then kill Claudius, why is his consciense getting in the way, especially if it was expected to revenge murders?