if she would like to marry Paris. He is an eligible bachelor who is of the same social and economic status as Juliet and would make a good match, at least in Juliet's parent's eyes. The Nurse is also impressed with Paris and she tells Juliet, 

A man, young lady—lady, such a man
As all the world—why, he’s a man of wax.

I’ll look to like, if looking liking move.
But no more deep will I endart mine eye
Than your consent gives strength to make it fly.

In order to understand why Romeo killed Tybalt, one must first address why Tybalt killed Mercutio.

Mercutio was not supposed to have been slain by Tybalt. Tybalt wanted to avenge himself on Romeo for crashing the Capulet's ball, which Tybalt felt to be an insult. In fact, Tybalt believed that the only reason why Romeo was there was to laugh at Tybalt's family. When Capulet stops Tybalt from fighting Romeo that night, Tybalt feels further insulted and vows to avenge himself, as we see in the lines, "I will withdraw, but this intrusion shall, / Now seeming sweet, convert to bitt'rest gall" (I.v.96-97). Mercutio learned that Tybalt had challenged Romeo to a dual by letter, and, feeling angered by this, followed his impetuous instincts and decided to place himself out on the street so that he could challenge Tybalt himself if he saw him.

Tybalt tries not to engage Mercutio in a fight, even addressing him and Benvolio politely as "gentleman" (III.i.38). But when Romeo tries to back out of Tybalt's challenge, because Mercutio is so angry with Tybalt, Mercutio considers Romeo's actions to be "vile submission!" and challenges Tybalt himself.

Hence, one reason that Romeo kills Tybalt is to avenge his best friend's death. However, the reasons go a bit deeper than that. Romeo, a lot like Mercutio, is also a very rash, emotionally driven, impetuous, and very young man. Tybalt ran away after he stabbed Mercutio. During that moment, Romeo had the opportunity to run from further calamity as well, allowing the Prince's law to punish Tybalt. Instead, due to his rash emotionalism, Romeo decides to break further laws by avenging his best friend's death himself.

Thus, we see that the reason why Romeo killed Tybalt is twofold: 1) He wanted revenge for the murder of his best friend; 2) His rash, emotional, impetuous mentality led him to believe that murdering Tybalt himself was the only way to avenge Mercutio's death.

Tybalt's frustration with Romeo in Act III, Scene 1 can actually be explained by something that occurred earlier on in the play. When Tybalt refers to the "injuries that [Romeo] hast done," Tybalt is referring to what occurred at the Capulet's ball.

At the Capulet ball in Act I, Scene 5, Tybalt recognizes Romeo's voice when Tybalt hears him asking a servant who Juliet is. Tybalt is furious to see that Romeo, a Montague, has invaded his family's private ball. Tybalt makes haste to fight with Romeo and to "strike him dead" for the "honour of his kin." However, Tybalt is actually stopped by Lord Capulet. Lord Capulet does not want bloodshed at or near his party, but he also rationalizes that Romeo is reported throughout Verona "to be a virtuous and well-govern'd youth." Therefore, Lord Capulet instructs Tybalt to leave him alone. Tybalt argues with his uncle, but then finally gives in. However, his uncle's treatment infuriates Tybalt even further. Tybalt feels he has been insulted by Romeo in his own house and promises to revenge himself on Romeo soon.

Hence, by the time we get to Act III, Tybalt feels he has been insulted by Romeo for two reasons: 1) Romeo invaded Tybalt's home and family; 2) Tybalt was put down by his uncle. Thus, a frustrated Tybalt also feels that Romeo is responsible his uncle's rebukes, which he feels were insulting.

Act III Scene 3 of Romeo and Juliet by William Shakespeare is set within the cell of Friar Laurence and occurs after Romeo has killed Tybalt. The main conflicts are over what Romeo should do. The Prince has decreed that Romeo be banished and Romeo rather than be grateful for this comparatively mild sentence rather melodramatically says that it would be better to commit suicide than to be separated from Juliet. The first conflict is with Friar Laurence, a Roman catholic monk, for whom suicide is a mortal sin. He tries to convince Romeo that banishment is better than death, but Romeo says:

“There is no world without Verona walls.” 

The Nurse enters and argues with Romeo that he owes it to Juliet to behave more responsibly, saying:

“Stand up, stand up! Stand, and you be a man.
For Juliet’s sake, for her sake, rise and stand!"

These conflicts are resolved when the Nurse and Friar Laurence devise a scheme for a secret marriage.

In the first scene of the play, a street fight breaks out between some boys from the house of Capulet -- Gregory and Sampson -- and some of Montague -- Abram and a nameless other.  Sampson bites his thumb at Abram, an insult, and they begin to quarrel.  After Sampson says that he serves as good a man as Abram and Abram asserts no better, Gregory sees Benvolio, a Montague, coming. He instructs Sampson to declare that his master is better than Abram's.  They draw their weapons and begin to fight, all but Benvolio.  Tybalt, Juliet's cousin, enters the scene and threatens Benvolio, who professes that he's only trying to keep the peace.  Just as it seems that the leaders of the families will get involved, Prince Escalus arrives and breaks up the fight.

Beginning the play with such a conflict, especially a conflict over something so stupid and minor, sets the stage for the violence that ensues.  We can see that the feud doesn't consist of anything beyond family pride -- perhaps no one even remembers what caused the initial disagreement.  The fight is ridiculous and especially frustrating to the prince who has such trouble maintaining order with such animosity in Verona.