Yes, you can often omit repeated elements when their meaning is clear. In a case such as this, it is common to replace them with commas:
Throughout the play, numerous characters evoke this central theme in their own unique way, such as Hamlet through his madness, Claudius, his ‘murder most foul,’ Polonius, his hypocrisy, Ophelia, her impurity, and Gertrude, her incestuous behavior.
The profusion of commas is a bit confusing, so we can use semicolons to separate the listed items:
Throughout the play, numerous characters evoke this central theme in their own unique way, such as Hamlet through his madness; Claudius, his ‘murder most foul’; Polonius, his hypocrisy; Ophelia, her impurity; and Gertrude, her incestuous behavior.
Some people simply omit the repeated elements without using commas:
Throughout the play, numerous characters evoke this central theme in their own unique way, such as Hamlet through his madness, Claudius his ‘murder most foul,’ Polonius his hypocrisy, Ophelia her impurity, and Gertrude her incestuous behavior.
That being said, I find nothing wrong with the original version and would probably prefer it over the shorter versions.