Why is the idea of playing a role or acting a part so important to Hamlet over the course of the play? How does role playing effect several major events in the plot and the relationships between various characters?

Hamlet was role playing all throughout the book. He was basically acting crazy the entire time so that he could avenge his father's death. Remember when he told his friends not to tell anyone if he acted a little weird after he saw the ghost? That was Hamlet's way of expressing that he was going to be acting crazy in order to avenge King Hamlet's death. So, basically, it's important to him because he needs to act crazy in order to avenge his father's death. He uses his insaneness as a sort of a crutch to commit all of the crimes without getting in trouble. So, one of the events it affects is the death of Polonius. Remember when Hamlet stabbed his sword through the tapestry and killed Polonius because he thought it was Claudius? That was definitely a major event affecting the play. That's when King Claudius started realizing Hamlet was just a tad bit dangerous and toyed to send him off to England. As for the relationship between the characters, think about his relationship with Ophelia. Ophelia was in love with him and his role playing teared them apart. It made Polonius tell Ophelia not to converse with Hamlet. And, also, look how Hamlet's insaneness turned Ophelia into a mad woman and made her basically commit suicide. There are quite a few other aspects that his role playing affected, but that should give you a start. He acted crazy all throughout the book, so just think off all of the events he caused by his lunacy.