Final answer:
The 'he's' in the quote refers to Hamlet; it's Polonius speaking about his plan to eavesdrop on Hamlet's conversation with Queen Gertrude by hiding behind an arras.
Step-by-step explanation:
In the given quote, "My lord, he's going to his mother's closet. Behind the arras I'll convert myself/To hear the process," the "he's" refers to Hamlet. This line is spoken by Polonius who is planning to hide behind the tapestry (commonly referred to as an arras) to eavesdrop on the conversation between Hamlet and his mother, Queen Gertrude. There’s a particular strategy Polonius employs throughout the play, which is listening in on others to glean information, and this quote is a direct representation of that approach.
The given dialogue showcases the dramatic intrigue and the dynamics of espionage that are central to the plot of Shakespeare's play, Hamlet.