There is no real evidence that would make us say that Romeo hates Tybalt personally as opposed to just hating the Capulets in general.
The two of them never actually speak to each other until the scene where Tybalt kills Mercutio and Romeo kills Tybalt. In this scene (Act III, Scene 1) Romeo doesn't say anything angry to Tybalt when Tybalt appears. Instead, he speaks soothingly to him. This implies he doesn't hate Tybalt. He does kill Tybalt, but in revenge for Malvolio.
So I don't see anything to imply that Romeo actually hates Tybalt.
There is no doubt that in the William Shakespeare play, Romeo and Juliette that Romeo and Tybalt have conflict. However, there is not a lot of evidence to support the idea that Romeo hates Tybalt. Romeo has definite problems with maturity and decision making--this is displayed throughout the play. but throughout their interactions Tybalt is the aggressor. Tybalt is the one who starts the fights--including wanting to attack Romeo when he shows up at the Capulet’s party uninvited. It is only the strong words and even stronger hand of Capulet that prevents Tybalt from attacking. When Romeo and Tybalt meet up later it is once again the actions of Tybalt when start the fatal confrontation. Romeo chooses to not walk away, but it is out of anger not hate, and he seems to feel true remorse when his actions result in the death of Tybalt.
Tybalt is the aggressor in the relationship between Romeo and Tybalt. Through-out the book Tybalt appears to be more hotheaded than Romeo. As in the previous answers, he is the one who starts the issues such as his anger at the party. Romeo does not become aggressive with Tybalt.
Vengeance creates a form of hatred fueled by loss and anger in Romeo's case. In this instance, following ' killing of Mercutio, Romeo feels compelled to avenge his friend's death. He could be avenging it in name only or to justify the wrong doing or he could be reacting in the strong emotion of hatred fueled by raw emotion.