Which two characters does Romeo kill in "Romeo and Juliet"?

While Romeo is the hero of Romeo and Juliet, he is a a tragic hero, which means that he does indeed commit acts that a romantic hero would not commit. Romeo does kill two young gallants, young noblemen all, one of whom roams the streets vexing the Prince because they insist upon venting the ire spurred by the family feuds surrounding them. An interesting question to accompany "Whom does Romeo kill?" is "What moral guilt Romeo bear for killing within the crucible of a family sanctioned feud?" The Prince seems to have this question for himself resulting in the royal edict that complicates Romeo's life in a way that leads to tragic ends for Romeo and Juliet and for all the mourners over their deaths.

In Act III, scene i, Benvolio warns Mercutio that, with "the Capulets abroad," they will not be able to escape a brawl with them. This foreshadows what is to come while also building suspense and developing the plot to bring in a tragic complication to the central conflict involving Romeo and Juliet. The debate Mercutio initiates as to Benvolio's character traits (given to false worry, hot tempered, and given to fighting)

MERCUTIO
    Thou art like one of those fellows that ... claps me his sword upon the table and says 'God send me no need of thee!'

    ... thou art as hot a Jack in thy mood as any in Italy, and as soon moved to be moody,

is a misdirected description of himself, as we see when, shortly following, Tybalt and his companions--of and loyal to the house of Capulet (feuding with and in opposition to the house of Montague)--enter and address the anxious Benvolio and the temperamental Mercutio who antagonizes at every opening between words so that Tybalt can neither ask his question nor keep his temper, being as hot tempered as Mercutio himself, as we saw at the ball when he would have drawn on Romeo when he was recognized as a Montague.

At this juncture, Romeo enters. Shakespeare has used Mercutio to build a fiery temper between the Capulet and Montague sides of the feud that is brought to a frothing frenzy when Tybalt calls Romeo a villain and that leads to Romeo's first slain victim. Romeo responds to Tybalt with love that Tybalt little understands and certainly in no degree accepts:

    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.

Tybalt responds by harkening to his remembrance of Romeo's mocking presence at the Capulet ball and responds by demanding that Romeo draw his sword and duel with him, presumably, in Tybalt's mind, to the death: "this shall not excuse the injuries / That thou hast done me; therefore turn and draw." Mercutio is incensed when Romeo shows "vile submission" by disregarding the challenge Tybalt has thrust down before Romeo to incite a duel. Mercutio rebukes Romeo for his response and, in his turn, challenges Tybalt to a duel. Sadly, the equally hot-headed Tybalt draws. Romeo dashes into the fray crying for Benvolio's help to separate the dueling Mercutio and Tybalt. In the confusion of swords--some fighting, some trying to "beat down" weapons--Mercutio is slain by Tybalt and, before he dies, curses both the houses of Capulet and Montague. Romeo, upon the return of Tybalt, swears vengeance upon him and they draw on each other, making Tybalt the first man Romeo slays:

ROMEO
for Mercutio's soul
Is but a little way above our heads,
Staying for thine to keep him company:
Either thou, or I, or both, must go with him.   

The Prince's edict banns Romeo to exile, rather than to have him slain as Tybalt slew first, which leads to the circumstances resulting in Romeo's second slaying of a man. Friar Laurence's plan to reunite Romeo and Juliet and to save Juliet from a polygamous marriage to Paris involves Romeo coming out of exile to meet Juliet at her would-be tomb in time to save her from real death as she rouses from virtual death.  

In Act V, scene iii, Paris, grieving at what he believes is Juliet's everlasting tomb, sees Romeo attempt to enter and, recognizing him as a loathed Montague, challenges him, denying him entrance. Romeo, of course, is adamantly persistent upon his entry, so he and Paris draw and fight. Paris falls as Romeo's sword slays him, making Paris the second man Romeo slays.

    PARIS
        O, I am slain!
        Falls
        If thou be merciful,
        Open the tomb, lay me with Juliet.
        Dies