Claudius rushed to marry Gertrude shortly after Old Hamlet's death, raising suspicions about his role. Throughout the play, Claudius remains suspicious of Hamlet and grows increasingly concerned about what Hamlet knows. Claudius' actions, such as staging the play to test Hamlet's reaction and making plans to send Hamlet away, seem to confirm Hamlet's belief that Claudius murdered his father to seize the throne.

2.  Claudius did not even wait one week after Old
Hamlet’s death to marry Gertrude and take over the
throne. A good leader would wait more then 1 week
before marrying the royal widow
 As described from Act 1 Claudius waited merely days
before marrying Gertrude.

3.  First off Claudius doesn’t address Hamlet in his throne
speech until the very end. As seen in Act 1 Scene 2 at
the end of Claudius’s long throne speech “How is it
that the clouds still hang on you?” Claudius did not
address the prince until the very end.
 Throughout the next acts Claudius is very suspicious
of Hamlet and his actions. Claudius then tells us in Act
3 how he is concerned about what Hamlet knows and
makes plans to send him to England.

4. Hamlet believes that Claudius murdered his father. That
fact is confirmed after the staged play that Hamlet sets
up in Act 3.
Lead by example, not killing to get what you want....

5.  He puts off a response to Fortinbras on Norway
passing through Denmark “It likes us well, and at our
more consider’d time we’ll read...”
 Diplomacy and state matters should come before
anything else.