Why are the king and queen worried about Hamlet?

The queen, Gertrude, is worried about Hamlet at the beginning of the play because Hamlet is still clearly mourning the loss of his father. She tells him,

Good Hamlet, cast they nighted color off,And let thine eye look like a friend on Denmark.Do not forever with thy vailed lidsSeek for thy noble father in the dust.Thou know'st 'tis common. All that lives must die,Passing through nature to eternity.  (1.2.68-73)

She...

The queen, Gertrude, is worried about Hamlet at the beginning of the play because Hamlet is still clearly mourning the loss of his father. She tells him,

Good Hamlet, cast they nighted color off,
And let thine eye look like a friend on Denmark.
Do not forever with thy vailed lids
Seek for thy noble father in the dust.
Thou know'st 'tis common. All that lives must die,
Passing through nature to eternity.  (1.2.68-73)

She asks that he stop wearing his black, mourning clothes and be friendly to the new king, his Uncle Claudius (who she calls "Denmark"), now his stepfather. She says that he cannot spend his life with his eyes cast down to the ground as though he were looking for his dead father there. He should know that it is common to lose one's father. Further, everything that lives eventually dies as that is the way of the world. In other words, she seems to say, enough is enough.

Claudius, the new king, and Hamlet's uncle and stepfather, professes his concern for Hamlet for the same reason but he seems somewhat less sincere. He claims, for instance, that "to persever / In obstinate condolement is a course / Of impious stubbornness" (1.2.92-94). In other words, then, he calls Hamlet obstinate, impious, and stubborn because the poor man mourns for his father, who died suddenly and fairly recently. As we learn later, Claudius is more worried about the risk Hamlet poses to him than he is for Hamlet's own welfare.