Before dying, Romeo speaks of the bodies of Paris and Tybalt, what do his words show about his personality at this point in Romeo and Juliet?

Romeo's address to both Paris and Tybalt shows Romeo's willingness to commiserate with his enemies about fate and death in addition to reiterating his passion for friendship and love.  I am reminded of Romeo's attempt to end the fray with words before being coerced into the final fight with Tybalt.  Romeo, then, stays true to his personality to the end.  In the tomb, first Romeo talks to Paris as Romeo himself lays Paris in the tomb.  Paris, right before he dies, asks Romeo to place Paris' body by Juliet.  This is the meat of Romeo's reply:

In faith, I will. Let me peruse this face. / Mercutio's kinsman, noble County Paris! / . . . [Mercutio] told me Paris should have married Juliet. / . . . I bury thee in a triumphant grave. / A grave?  O, no, a lantern, slaughtered youth, / For here lies Juliet, and her beauty makes / This vault a feasting presence full of light. Death, lie thou there, by a dead man interred.  (5.3.74-87)

Romeo, then, does Paris a great kindness:  Romeo lays Paris by the "lantern" of his love that will light Paris' tomb.  Romeo then notices how fair Juliet looks (making the audience wish like hell she would go ahead and wake up already), but immediately turns to Tybalt:

Tybalt, liest thou there in thy bloody sheet? / O, what more favor can I do to thee / Than with that hand that cut thy youth in twain / To sunder his that was thine enemy? / Forgive me, cousin!

Romeo, then, also does Tybalt a great kindness:  he kills Tybalt's most loathed enemy (Romeo) and calls Tybalt "cousin."  Any audience member who has seen the deep friendship that existed between Mercutio and Romeo should have expected nothing less from him here.

In regards to his personality, Romeo is of such a melancholy humor that he commiserates with his dead enemies in a discussion about their "sour" fate.  It shows that, in the face of death, there is no reason for hate.  (Although, I had to laugh when I found that Romeo devoted a full 13 lines to Paris and only 4.5 to Tybalt.)  This mirrors the end to the feud that is revealed at the culmination of the play.  This episode also shows Romeo's continued admiration for Mercutio, since he is so highly spoken of in Romeo's speech here.  Finally, as Romeo speaks to Paris, his vast love for Juliet is still evident as it has been from the very first moment he saw her.  Love, of course, is Romeo's true calling.  Furthermore, Romeo has never departed from these aspects of personality and, even though he is prone to mood swings, has remained steadfast throughout the play.