How does Claudius manipulate Rosencrantz and Guildenstern in the play?
In addition to hiring Rosencrantz and Guildenstern to help him with Hamlet's depression, Claudius perhaps has planted the seed that Hamlet's melancholy has resulted from the fact that his ambition to inherit the throne has been thwarted by Claudius and Gertrude's marriage. Notice in Act 2 how often Rosencrantz and Guildenstern use the word "ambition" in their questioning of Hamlet. It seems here that Claudius has commissioned Rosencrantz and Guildenstern not only to find out what is wrong with Hamlet, but also to determine if Hamlet had past ambitions for the throne, and if he is still harboring those same ambitions. Claudius obviously views Hamlet as a true threat, and he is using the two spies in an attempt to find out Hamlet's motivation.
Later, Claudius uses Rosencrantz and Guildenstern to escort Hamlet to England where Claudius plans for him to be executed. Of course, this plan does not work, but we are not sure of Rosencrantz and Guildenstern's knowledge of this plan. Perhaps they are innocent of the orders they are carrying. Perhaps not. Hamlet likens them to "sponges" or "knee-crooking" knaves--sychophants that will do whatever they are told to promote their standing with the king.
The short answer would be: very easily. Rosencrantz and Guildenstern are not the brightest guys around, but Claudius calls them to Denmark ostensibly because he is worried about Hamlet and he is hoping that they will be able to help him figure out what is wrong and presumably fix it.
As they arrive, Hamlet sees through their entrance as they cannot but admit that they were sent for, they did not in fact come of their own accord.
Claudius continues to use them by stirring up their sense of duty to royalty as well as that to their friend Hamlet and the idea that he is sick or somehow out of kilter and that together with Claudius if they try hard enough they will be able to help him escape his melancholy, etc.