In Hamlet, is there any interplay between how individuals perceive themselves and how they are perceived by others?

Yes. The idea of acting and perception are key themes in Hamlet and many other plays of Shakespeare's. The play revolves around Hamlet, the young prince. He continually deceives others and himself in his efforts to find his moral center and avenge his father.

For example, early in the play the other characters misunderstand Hamlet's strange behavior. He is deeply upset by his father's death and his mother's quick marriage. The courtiers falsely believe that this is because of his relationship with Ophelia. Because of this mistake, Claudius and others perhaps underestimate the threat he posses until later in the play. Additionally, the disregard for his true feelings alienates him even further from his mother and Ophelia.

Even more clearly, Hamlet later fakes madness in order to deceive his enemies. This act deeply changes Hamlet's character and leads some experts of the play to even conclude that the act eventually drives the prince to genuine insanity.

A final example would be the play that Hamlet stages to confirm his uncle's guilt. His uncle appears normal in some respects, but Hamlet believes the emotions of the play will lead Claudius to break the act and reveal his true feelings. The fiction of the play reveals the truth of the king's feelings. 

This continuing theme of acting, lies, and false perception is a key theme in Hamlet. It explores the relationship between fiction and reality.