What is the king's facade in his speech in Hamlet?
Because you don't make reference to a specific speech, I will discuss his speech at the start of Act 1 Scene 2 when we meet Claudius for the first time. Much of how he portrays himself serves as the foundation of his character from the rest of the play.
The first speech is Claudius's address to the court. In the first third of the speech, Claudius puts on the facade of the loving brother who is still mourning his dear recently deceased dead brother. At this point in the play we don't know yet that he is the murderer of King Hamlet. In the next third of the speech he addresses the issue of his quick marriage to his former sister-in-law Queen Gertrude. He acknowledges that the situation is potentially awkward, but thanks the court for "going with this affair along." He is putting on the facade of pretending he cares what they think of his actions. The last third of the speech addresses the foreign policy/external threat posed by young Fortinbras. He puts on the facade of an unflappable, strong leader. He suggests that he has the situation under control and rather flippantly says, "so much for him" as if to mean that Fortinbras is merely a pesky fly, but a legitimate threat to the nation. Clearly, Claudius is playing the game well of hitting all the right notes with this audience.
In the next speech he is directly speaking to Hamlet about Hamlet's prolongedmourning of his father. He is putting on the facade of loving and concerned step-father. He attempts to be practicial and logical in regards to the death of Hamlet's father. He points out that everyone dies eventually and that all children must bury their fathers at some point. He suggests that this mourning is a "fault to heaven" perhaps knowing that Hamlet is rather religious. He may be putting on the facade of being helpful, but both Hamlet and the audience see his comments as rude and insulting.
In almost all of Claudius's speeches, he is putting on the facade of being "kingly" and for looking out for Hamlet's best interests, but in actuality, he is only looking to maintain himself as king and rid himself of the threat of Hamlet. You can see this in his conversations with Rosencrantz and Guildenstern in Acts 2 and 3, as well has his long scene with Laertes in Act 4.