How did Romeo feel about Killing Tybalt in Romeo and Juliet?

Romeo feels terrible about killing Tybalt, because he was Juliet’s cousin and he didn’t want to fight him.

The death of Tybalt is a turning point in the play.  The fiery tempered Tybalt has been targeting Romeo ever since Juliet’s party, where he felt that Romeo’s presence there was an insult to the Capulets.  Tybalt tracks down Romeo and tries to fight him in the street, but Romeo refuses.  As far as he is concerned, Tybalt is family.

TYBALT

Romeo, the hate I bear thee can afford
No better term than this,--thou art a villain.

ROMEO

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting: villain am I none;
Therefore farewell; I see thou know'st me not. (Act 3, Scene 1)

Romeo is not getting rid of Tybalt that easily.  He is stubborn, and when Romeo does not fight him he fights Mercutio.  Mercutio tries to protect Romeo, but Romeo tries to stop the fight.  He gets in the way, and Mercutio is killed.  Now Romeo has no alternative but to fight Tybalt.  He has to avenge Mercutio’s death, and there is no one left to fight for him. 

BENVOLIO

Romeo, away, be gone!
The citizens are up, and Tybalt slain.
Stand not amazed: the prince will doom thee death,
If thou art taken: hence, be gone, away!

ROMEO

O, I am fortune's fool! (Act 3, Scene 1) 

Other than the fact that he killed his wife’s cousin, Romeo has a bigger problem.  Prince Escalus decreed that anyone fighting in the streets would be put to death. The prince takes pity on Romeo and has him banished instead, but to Romeo that is a fate worse than death.  He can’t stand to be without Juliet. 

Romeo leaves, but he has a very rough time with the concept.  He returns to find Juliet supposedly dead in her tomb.  Friar Lawrence tried to get him a message, but it did not make it through.  Romeo is so devastated that he kills himself with poison.  Juliet wakes up, realizes what happens, and stabs herself.