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.