Question

In what sense is Hamlet wrong in his plotting against and killing of Claudius?

Answer 1

He is wrong in killing Claudius because killing is wrong unless it is self defense.

Answer 2

He can't justly kill Claudius unless he knows Claudius is guilty. The fact that Claudius is guilty is not enough by itself.  You can't gamble on the guilt of another by executing them: the gamble is itself tantamount to depraved indifference. You need proof. Hamlet is clear on this: he won't gamble, since the spirit he saw may be a devil.