1. Conniving. He constantly manipulates characters around him to serve his own means. He pulls off a sneaky murder of Hamlet's dad, manages to woo and win the queen, convinces Rosencrantz and Guildenstern, who were buddies with Hamlet, to spy on Hamlet and then try to murder him, and then convinces Laertes, through some pretty tricky manipulation, to challenge Hamlet to a fencing duel that has the added dash of excitement by throwing poison into the mix.
2. Ambitious. Most likely to gain control and power in the kingdom by being king, he murders the king and marries the queen. And, he is willing to protect the fact of his murder at all costs, even if it means dispensing of his stepson. His ambition and drive to be king outweighs even his sometimes troublesome conscience.
3. Thorough. He goes to great lengths to watch Hamlet. He is very cautious and wary about Hamlet's behavior. He is worried Hamlet might know, and so goes to lengths to figure out if Hamlet truly is mad, or if he really knows what happened. He is very thorough in having Hamlet analyzed; he is in on the whole Ophelia set-up, enlists Hamlet's buddies to help him out, and makes sure that Hamlet is dispatched quickly when troubles arise. He covers all of his bases, and leaves no stone unturned in making sure his secret is hidden.
I hope that those words help to get you thinking; good luck!
Of the many words that could be used to describe Claudius, the best might be the one that is used several times by Hamlet himself after his interview with his father's ghost. Hamlet says:
O villain, villain, smiling, damned villain!
My tables--meet it is I set it down
That one may smile, and smile, and be a villain;
Shakespeare is giving instructions that the actor playing Claudius should be constantly smiling. He smiles to conceal his villainous motives and intentions. He smiles because he is pleased with his success. He smiles to show everyone what a kind a gracious monarch they have acquired. He smiles to hide his guilt and fears. Even in tense situations Claudius would act perfectly at calm, reasonable, and friendly. He would have discovered that a smile can be the most effective way of dealing with practically everybody in every sort of situation. For instance, when Hamlet is brought before him after killing Polonius, Claudius would not roar and threaten but would treat him the way a loving father might treat a naughty boy. He would smile and say in a confidential tone, "Now, Hamlet, where's Polonius?" Claudius would pride himself on his coolness, his "grace under pressure," to use Hemingway's phrase. Even though he is a treacherous murderer, Claudius shows himself to be a very competent king. This may be one of the reasons that Hamlet seems reluctant to assassinate him; he feels that Claudius makes a better king than he would make himselff. (Malcolm seems to have similar feelings about Macbeth which he expresses to Macduff in that play.)
So one good word for Claudius would be "smiling." Not all villains--in plays or in real life--are smiling villains. Another good word, of course, would be "villain." I would suggest that a third good word would be "drunkard." Claudius is usually shown drinking wine in large quantities. This is not because the liquor makes him happy but because he has a terribly guilty conscience. He not only feels guilt and remorse, but he feels that, like Macbeth, he has condemned himself to eternal damnation. Furthermore, he is eaten up with fear. Having obtained the crown by murder, he realizes that someone else could obtain it from him by the same means. He suspects that that person could be Hamlet, since his nephew has the most obvious motivation along with a genuine right to be king. Throughout the play Claudius is trying to pry into Hamlet's mind, keeping him a virtual prisoner and under continuous surveillance, using Polonius, Ophelia, Gertrude, Rosencrantz, Guildenstern, and no doubt many other courtiers and household servants to spy on the Prince.
In Shakespeare's "Hamlet," because Claudius is treacherous and shrewd and unethical, Hamlet refers to their battle of intelligence and will as that of "mighty opposites":
'Tis dangerous when the baser nature comes/Between the pass and fell incensed points/Of mighty opposites. (V, ii, 60-62)
Although Claudius is able to handle competently and shrewdly the threats to Denmark by Fortinbras of Norway, and send Hamlet off for a while, when troubles brew; clearly, Claudius is the "something [that] is rotten in Denmark." He pits Rosencrantz and Guildenstern, Hamlet's friends, against Hamlet. He causes Hamlet's mother to fall from favor with her son, he manipulates Laertes into dueling with Hamlet. In his "confession" scene, although Claudius admits that his "offence is rank, it smells to high heaven" (III,iii,36), he continues his hypocrisy:
O wretched state! O bosom black as death!/O limed soul, that struggling to be free/Art more engaged! (III,iii,67-69)
More "engaged" in the final act, Claudius is forced by Hamlet to drink the poison from the cup intended for the prince so that he may treacherously remove any competition for his crown.