A good example of the contrast between Claudius's smooth facade and his crimes is Act IV, scene iii of Hamlet in which Claudius' double-dealing nature is immediately evident. Polonius has just been murdered. King Claudius is alone (except for his attendants) and speaking to himself out loud in a short speech, which doesn't really qualify as a soliloquy.
Claudius is saying that Hamlet is "dangerous" and must not be allowed free reign in the castle: "How dangerous is it that this man goes loose!" Claudius is weighing the wisdom of not treading harshly against Hamlet as he is beloved in Denmark and harsh treatment would redound against himself, Claudius. He decides upon dissembling and appearing to "bear all smooth and even" to avoid causing a revolt. He will send Hamlet away--to all appearances--speedily yet gently:
This sudden sending him away must seem
When hamlet enters, Claudius does an about-face, so to speak, and expresses nothing but solicitude for Hamlet's well being. Hamlet of course replies with bitter witticisms:
Where is Polonius?
At super! where?
Not where he eats, but where he is eaten: ....
After pinpointing where Polonius's body lies, Claudius dissembles and expresses great concern for Hamlet, for his "espcial safety-- / ... as we dearly grieve / ... must send thee hence / With fiery quickness ..."
Once Hamlet has left "For England!" Claudius reveals his intention of having England (a metonymy for the King of England and his lieges) kill Hamlet the moment he alights at the end of his journey--England must comply with this demand in order to continue the treaty of peace that tribute payments have secured with Denmark: “and thy free awe / Pays homage to us.”
This scene shows that Cluadius's smooth facade has behind it the heart of a man who is unethical and immoral; who is the "monster" Hamlet declares him to be; who is manipulative; and who is an eloquent "villain." The facade derives from the skill of appearing to be what he is not. It is thus that Claudius's character best depicts Shakespeare's universal theme of reality versus appearances in which Claudius's behind-the-scenes actions like his crimes of murder and usurpation are the reality, while the facade of his kingly grace and loving devotion to his queen is but the shadow of appearance.