Claudius is an inherently devious and manipulative character. Having carefully plotted and schemed his way to the top, he is suddenly confronted by an unexpected problem: Hamlet's desire for revenge, spurred on by the ghost of his murdered father. Claudius needs to use every ounce of his manipulative powers to stop Hamlet from getting any ideas in the direction of acting on his desire. At the very least, he needs to encourage his crestfallen nephew to move on and cease his endless moping around, his "unmanly grief":

'Tis sweet and commendable in your nature, Hamlet,
To give these mourning duties to your father:
But, you must know, your father lost a father;
That father lost, lost his, and the survivor bound
In filial obligation for some term
To do obsequious sorrow: but to persever
In obstinate condolement is a course
Of impious stubbornness (Act I Scene II).

Claudius's whole marriage to Gertrude stands as a testimony to his guile and manipulation. And this little speech, though ostensibly addressed to Hamlet is really for Gertrude's benefit. Claudius wants to get Gertrude to persuade Hamlet not to go to university in Wittenberg; in doing so, he is playing on her motherly affections. Claudius wants Hamlet where he can see him, the better to be manipulated and controlled.

Or rather Claudius wants someone else to keep an eye on Hamlet, namely Rosencrantz and Guildenstern. Claudius would have them (and us) believe that he is merely looking out for the boy, worried by his growing gloom and melancholy. In fact he is actually trying to feel Hamlet out to gain a sense of his ambition in relation to the throne. It is noticeable just how often Rosencrantz and Guildenstern use the word "ambition" when they talk to Hamlet.

Later on, Claudius manipulates Hamlet's school friends further when he sends them on a mission to accompany Hamlet to England. Hamlet is considered mad and must leave Denmark on the grounds of public safety. But in truth Claudius has other ideas. Rosencratz and Guildenstern have been duped into participating in a wicked assassination plot.

Contrast Claudius's approach to Hamlet with how he later on deals with Laertes. He is every bit as manipulative but this time for a wholly different, and more deadly purpose:

Not that I think you did not love your father;
But that I know love is begun by time;
And that I see, in passages of proof,
Time qualifies the spark and fire of it.
There lives within the very flame of love
A kind of wick or snuff that will abate it;
And nothing is at a like goodness still;
For goodness, growing to a plurisy,
Dies in his own too much: that we would do
We should do when we would; for this 'would' changes
And hath abatements and delays as many
As there are tongues, are hands, are accidents;
And then this 'should' is like a spendthrift sigh,
That hurts by easing. But, to the quick o' the ulcer:—
Hamlet comes back: what would you undertake,
To show yourself your father's son in deed
More than in words? (Act IV Scene VII).

Claudius wants Hamlet to stop brooding on the death of his father and move on. With Laertes it is the exact opposite: Laertes must never forget Polonius and should avenge his death as soon as possible. In both cases, Claudius is thinking, as always, of his own motives.

Concerning Shakespeare's Hamlet, in short, King Claudius manipulates the characters you ask about in the following ways:

• He gets Gertrude to marry him and, thus, gets the crown.  He also manages to keep Gertrude viewing Hamlet on his terms and from his point of view for most of the play:  at least until the bed chamber scene in Act 3.4, and possibly until the final scene in the play when she realizes Claudius has poisoned the cup meant for Hamlet.  Gertrude is constantly on Claudius's side and perceives every issue that concerns Hamlet from the same point of view as Claudius (that Hamlet should shake off his sadness, that he should stay at Elsinore and not go back to school, that Hamlet may be "mad" because he is in love with Ophelia). 
• Claudius gets Ros. and Guil. to spy on Hamlet for him.  He turns them from being Hamlet's friends to his own agents.  He summons them to Elsinore for this purpose and uses them for it.  He repeatedly sends them on errands to Hamlet:  to find out what's bothering him, for example, and to find out where Polonius's body is. 
• Claudius redirects Laertes's wrath over the death of Polonius away from himself and toward Hamlet.  He leads Laertes to think that he is helping him kill Hamlet for Laertes's sake, when in reality he is helping Laertes in order to protect his own crown. 

Concerning Hamlet, however, Claudius attempts to manipulate Hamlet, but it doesn't work out in his favor.  Claudius thinks he is playing a cat-and-mouse game with Hamlet, and he is--the only trouble is that Hamlet is the cat, not Claudius. 

Notice how much time and effort Claudius spends trying to figure out why Hamlet is "mad."  This is by design--Hamlet's design.  He tells Horatio that he will be putting on an "antic disposition" (Act 1.5.171)--pretending to be mad.  This is a diversion designed by Hamlet.  And it works.  Claudius spends all his time trying to figure out why Hamlet is insane, and never figures out--until Hamlet wants him to--that Hamlet knows about Claudius's murder of King Hamlet. 

Only when Hamlet gets what he considers to be proof or confirmation that the Ghost is telling him the truth and that Claudius is guilty (by watching Claudius's reaction to the murder scene in the play-within-the-play) does Claudius know Hamlet knows.  And notice what Hamlet tells Claudius the name of the play is--The Mousetrap.  Claudius doesn't manipulate Hamlet; Hamlet manipulates Claudius.  Hamlet is the cat.