What does claudius mean wh Though yet of Hamlet our dear brother's death The memory be green, and that it us befitted?

In this quote from Shakespeare's
Hamlet, found in Act 1.2, Claudius is justifying his own actions. 
He is explaining to his subjects why he has done what he's done.  Indirectly (from the
writer's point of view, not the character's), he is also outlining one of the central
issues of the early part of the play.

This scene is the
audience's introduction to the court of Elsinore.  When Claudius refers to King Hamlet's
death, he's informing the audience that the king of Denmark has recently died, in
addition to beginning his logical train of thought that justifies his hasty marriage to
the dead king's wife, Gertrude.  The memory is fresh, green, but people must continue to
lead their lives.  Thus, he married Gertrude.  The kingdom must continue to remember
King Hamlet, yet must also continue to live.

This logical
argument then leads into the new king's questioning of young Hamlet.  First, young
Fortinbras has to be taken care of, as well as Laertes.  But when those situations are
quickly handled, Claudius moves to Hamlet.  The logic is carried over:  everyone else,
though still respectfully remembering the deceased king, has recovered and moved on, why
can't young Hamlet?  Notice the logical chain:  Life must go on, situations like
Fortinbras must be handled; thus, Claudius married Gertrude.  Look how well Claudius is
handling things!  Everyone else is doing it--why can't young
Hamlet?

Here's what he
says:

But now,
my cousin Hamlet, and my son--....

How is it that the
clouds still hang on
you?

When everyone else has
recovered, how is it that you are still upset?

And the play
is on!  Gertrude's hasty remarriage is a key issue in the play, and an event that Hamlet
obsesses about throughout the play.  Claudius attempts to justify it in this scene, and
uses his logical argument to convince Hamlet that he, too, should go along with
it.

His argument fails.