When we first meet Tybalt in Act 1, scene 1, he comes across as an impetuous, violent character who spews hatred and disgust for all the Montagues. His insults are a deliberate ploy to draw Benvolio into a fight. It is ironic that Benvolio should be so easily drawn into battle when, as he says, "I do but keep the peace." The fact that Benvolio does respond by fighting is an indication of how easy it is for Tybalt to manipulate others to do his will. There are, obviously, other factors that drive Benvolio to fight, but primarily, it is Tybalt's blatantly deliberate provocation that forces the issue.
The scene clearly indicates Tybalt's sly purpose: he wishes to create turmoil. He says he hates the word peace and would, therefore, ensure constant disruption. In addition, when the prince arrives, he and his cohorts disappear, leaving Benvolio to face his anger. Tybalt cunningly removes himself from the scene when it was he and his companions who started the fracas in the first place.
It is this particular attribute that makes Tybalt suited to the character he is named after. In Reynard the Fox, the cat named Tybert abandons Reynard's father when they see a group of hunters. Reynard's father relied on the cat for help, but Tybert climbed into a tree, cunningly leaving the father to face the hunters who then beat him up.
Tybalt's efforts at manipulation are not all successful, though. When he sees Romeo at the Capulet ball, he wants to punish him for daring to gatecrash their party. When he tries to incur his uncle's wrath upon Romeo he is stopped in his tracks. His uncle, Lord Capulet, is upset and commands him to ignore the young Montague and not start a scene. Tybalt is obviously humiliated and feels that he has to avenge Romeo's impudence.
He later challenges Romeo to a duel by sending a letter to the Montague household. This does not work, though. When he confronts Romeo, the latter refuses to get into a fight with him because he now loves Tybalt more than he would know since he has fallen in love and married Juliet, Tybalt's cousin. Romeo feels that Tybalt has become family and does not want to harm him.
Tybalt is not pleased, of course, and will have his way. When Mercutio takes up his challenge Tybalt is more than eager to indulge him. Tragically, Tybalt mortally wounds Mercutio when he spies an advantage during Romeo's attempt to stop the duel. This incident also fits the description of Tybalt as a cat for he is aware and quick to take advantage of a situation, just as Tybert does in Reynard the Fox. Furthermore, Tybalt is nimble and quick, just as a cat is - a fact alluded to by Mercutio when he says that Tybalt is:
More than prince of cats ...He fights as
you sing prick-song, keeps time, distance, and
proportion; rests me his minim rest, one, two, and
the third in your bosom:
Tybalt's actions have tragically serious repercussions for it is his actions that sets in motion a series of events which end not only in his untimely demise but also that of our two star crossed lovers and the county Paris.
In terms of being sly, one of the ways that Tybalt is particularly sly is in the way that he can manipulate other characters to do his bidding. Without being particularly long winded or having the number of lines that other characters do, Tybalt has a major influence on the action of the play as he is able to push other characters to react the way he wants to.
He is also sly like a cat in that he is always looking for any advantage and never, ever hesitates to take it. If you look carefully at the fight that Tybalt basically engineered between him and Mercutio because he had been itching for revenge for the perceived slight the family had received, his slyness is revealed in spades. In the very famous scene of the fight between Tybalt and Mercutio, he sees an opening when Romeo steps in to stop the fight and immediately stabs Mercutio under Romeo's arm.
Tybalt is also particularly sly in his words as he uses them to great effect to draw out the reaction he is looking for in other characters.