When the character first appears
His first lines are:
"What, art thou drawn among these heartless hinds?
Turn thee, Benvolio, look upon thy death." Act I, Scene I, Lines 65-66.
It is already clear that Tybalt is heavily driven by the need to fight for his family. In his second line he challenges Benvolio to a fight. His motivation is to provoke conflict.
The picture above is taken from Baz Luhrmann's film of 'Romeo and Juliet'. In the film Tybalt is cat-like, sly and vain. When Tybalt enters the room there should be a change in the attitude of other characters. He should make other characters feel nervous and to some extent, fearful. In the photo above Tybalt stands confidently. This is how he should be in the play.
How he acts
What he says
What he thinks
The relationships he has with other characters
Mercutio and Tybalt have a strong relationship. They share a similar motivation, to defend what they care about. Tybalt wants to fight for his family, while Mercutio wants to defend his friends. Tybalt and Mercutio are also aggressive and easily angered. They are enemies and despise each other. It is these similar characteristics that drive them both to their death. When Tybalt challenges Romeo to a fight and he declines, Mercutio cannot help but fight for his friend.