Ambitious: He craved the throne, and went to great lengths to obtain--and keep--his position as king once he gets it. No one would murder and connive quite so much unless they were ruthlessly ambitious.
Manipulative: Claudius is able to manipulate other people to do what he wants them to. Take for example Polonius, who does his bidding all of the time. Also take Rosencrantz and Guildenstern; they used to be trusted friends of Hamlet's, but under Claudius' manipulation turn into spies and potential murderers. Then, Laertes is cunningly manipulated into challening and intending to kill Hamlet.
Selfish: Claudius takes what he wants, with no concern for others. He murders his brother and woos the queen, all to obtain the kingdom. Whether this hurts people in the process is of little concern to him.
Conflicted: One telling scene in the play shows Claudius on his knees, praying, tormented by his sins. So despite his crimes and cunning nature, he still has a conscience that afflicts him occasionally.
Clever: Even though Claudius is able to take the throne and manipulate people, he is very smart in how he does it. He is aware of his position and popular opinion, and works around it very well. Instead of having Hamlet killed for Polonius' murder, he sends Hamlet away on a ship, ordering his death away from the kingdom where people might rebel. He carefully arranges for Hamlet's death to look like it had no connection thi himself. He is smart, clever, and uses that gift to his benefit.
I hope that helped; good luck!