How is the interplay between how individuals perceive themselves and how they are perceived by others developed in Shakespeare's Romeo and Juliet?

There is, indeed, a direct relationship between how we see ourselves and how others see us. For example, if we wrongly see ourselves in a negative light, we soon begin to act upon our vision of ourselves, leading others to see us in a new, negative light as well. Similarly, if others wrongly insult us based on their own wrong or prejudiced assumptions of us, we soon begin to see ourselves in the same negative light they see us and to act upon this vision. Both of the above accounts are true with respect to positive perceptions as well. In Shakespeare's Romeo and Juliet, this same interplay between the perception of self and the perception of others leads to new realities.

One example can be seen in Romeo's perception of himself. In Act III, Scene 1, Romeo, who has just married Juliet, feels happy with himself and at peace with the world. He also sees the Capulets and Montagues amicably uniting as one family as a genuine possibility. More importantly, he sees himself as a good, caring, completely innocent person who is capable of bringing the two families into a peaceful union. Hence, Romeo is genuinely shocked when Tybalt, offended at seeing Romeo crash the Capulets' party, calls Romeo a "villain" and challenges him to a duel. Romeo's reply shows his effort to make Tybalt understand what a caring, peaceful person Romeo truly is.

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 III, Scene 1, lines 59-62).

In speaking of his genuine love for Tybalt, just as he now loves all Capulets as a family due to his love for Juliet, Romeo strives to show Tybalt he is a caring person. Also, in trying to walk away from Tybalt's challenge, Romeo strives to show Tybalt he is a peaceful person. In addition, Romeo maintains his view that he is an innocent person when he asserts, "I do protest, I never injured thee" (Act III, Scene 1, line 65).

Regardless of Romeo's attempts to alter Tybalt's perspective and pacify the situation, Tybalt refuses to see Romeo in any other light than how Tybalt wants to see him: a hateful scoundrel, which is how Tybalt sees all Montagues. Due to Tybalt's inability to see Romeo in any other light, Tybalt persists in challenging Romeo. When Romeo refuses to accept Tybalt's challenge, Mercutio attacks Tybalt instead. Mercutio is then stabbed to death by Tybalt when Romeo tries to break up the fight. At this moment, Romeo becomes exactly how Tybalt perceives him to be — he becomes a hateful, violent villain who seeks the revenge of his friend's death, which leads to Tybalt's murder, Romeo's exile, and other subsequent deaths.