Tybalt: This, by his voice, should be a Montague.

Fetch me my rapier, boy. What! dares the slave

Come hither, cover'd with an antick face,

To fleer and scorn at our solemnity?

Now, by the stock and honour of my kin,

To strike him dead I hold it not a sin.

-Romeo and Juliet,

William Shakespeare

How does Tybalt's relationship with Romeo affect his reaction to Romeo being at the party?

Tybalt holds a deep hatred/grudge against Romeo, as he is a Montague. He has fought with Romeo on the streets as well, so their relationship is tense. When Tybalt realises that Romeo must have snuck into the Capulets' party, he is enraged and does not hesitate to kill him.

So I guess, due to Tybalt's deep-rooted hatred for the Montagues and, therefore, Romeo, he loses his sense of morality and on impulse, decides to kill him.

He thinks Romeo is there to insult his family because of the ongoing feud.