There is a lot more to this quote than you've put here and the quote doesn't make a lot of sense if it ends right here. But I don't know how far you want to go in this speech.
In the part you put down here, all that Claudius is saying is that Hamlet has just recently died (his memory is still green). Then he says that it is appropriate for him (he uses the royal "we") to... that's where the quote ends.
In the play, Claudius goes on to say
To bear our hearts in grief, and our whole kingdom
To be contracted in one brow of woe,
Yet so far hath discretion fought with nature
That we with wisest sorrow think on him
Together with remembrance of ourselves.
This is what it's appropriate to do. He says it's appropriate for them to continue to mourn Hamlet senior. But it is also important for them to keep living their own lives. This, he says, is why he thought it was okay to go ahead and marry Gertrude so soon.
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.