What public and private morality does Claudius demonstrate in Act IV of Hamlet

Claudius as a king tries to protect and preserve the public morality included in the social order, ensuring that his nephew Hamlet must be punished for murder. His decision would have to be irrevocable as king, and he would have to treat Hamlet just like any other subject. However, being a murderer himself, he is afraid of Hamlet and what he might find out. Therefore, he is desperate to get rid of him, sends him off to England, and arranges for his murder, while the king inside him conflicts with the offender in him.