What is a good quote by Claudius in "Hamlet" that shows that he thrives on power?
When Claudius determines to send Hamlet to England, he tells Hamlet's old college friends, Rosencrantz and Guildenstern,
I like him not, nor stands it safe with us
To let his madness range. Therefore prepare you.
I your commission will forthwith dispatch,
And he to England shall along with you.
The terms of our estate may not endure
Hazard so dangerous as doth hourly grow
Out of his lunacies. (3.3.1–7)
Claudius is unhappy with Hamlet's behavior, and he says that it is too dangerous for Claudius to let Hamlet's mental condition remain so uncontrolled. He tells Hamlet's friends to get ready because he is going to send them to England on business, and they'll take Hamlet with them. In his role as king, Claudius says that he cannot risk Hamlet's lunacy, which is getting worse each hour.
This speech shows that Claudius is far more concerned with his own power than he is with his step-son/nephew's health (a fact that is confirmed when we learn that he's sent a letter with Hamlet's friends instructing the king of England to put Hamlet to death upon his arrival). Claudius is concerned not that Hamlet is a lunatic, but that his lunacy may reflect poorly on Claudius. This helps to show how much he does, indeed, thrive on power. He is willing to sacrifice anyone and everyone—including his brother, his wife, and his nephew—to gain or keep it.
In Act 3, sc. 3, Claudius goes to the chapel to try to pray. He realizes that Hamlet somehow knows that he, Claudius, killed his brother, Hamlet's father. The play that he and others just witnessed, depicted his crime, and so Claudius is in the chapel. About half way through his soliloquy in the chapel, Claudius asks if it's possible for him to be forgiven for his sin when he still reaps the benefit of that sin. "That cannot be, since I am still possessed/ Of those effects for which I did the murder:/ My crown, mine own ambition, and my queen." Even though he puts his ambition in the middle, it is essentially the same as the first reward he mentions, "..My crown..". The crown is something he wanted because of his ambition; he wanted to be king and like Macbeth, he was willing to kill to expedite his ascension to the throne. Further proof of his desire for the throne and the fact that the desire overrides his love for Gertrude is found in the last scene of the play. While Claudius does say to Gertrude before she drinks the poisoned wine, "Gertrude, do not drink." He does nothing else to stop her from drinking what he knows absolutely to be poisoned wine. Rather than expose what he's done, he is willing to let the supposed love of his life die. Clearly power and with it, the throne, are most important to Claudius.