Explain the significance of the play "The Mousetrap" as a metaphor for Hamlet's relationship with other characters in the play in Hamlet.
There are a number of aspects to "The Mousetrap" and how it functions, and I am glad that your question points towards its deeper significance rather than merely focusing on how it relates to Hamlet's realtionship with Claudius. Certainly, this would be one of the central aspects concerning the play within a play. Let us remember the famous lines that Hamlet gives in his soliloquy at the end of Act II:
I'll have grounds
More relative than this: the play's the thing,
Wherein I'll catch the conscience of the King.
The Mousetrap then becomes part of the elaborate game of cat-and-mouse that Hamlet plays to try and work out whether the Ghost's words are true before he does something irrevocable like kill Claudius and ends up in hell. Yet, at the same time, in a sense, The Mousetrap is a repeated motif that is echoed throughout the entire play with Hamlet himself as the mouse. Note the number of times in which Hamlet is eavesdropped upon and observed to try and ascertain what is wrong with him. Consider the number of times he is approached by someone who was formerly a friend or lover or caring family member with a very different agenda. Rosencrantz and Guildenstern, Polonius, Laertes, Ophelia, and his own mother are all used by Claudius to try and trick and trap Hamlet in various ways. Whilst we associate The Mousetrap with something that Hamlet does to Claudius, actually, he is merely repeating what Claudius has already done many times and will continue to do to Hamlet. Hamlet is merely returning the favour.
A recent production of the Shakespeare Company of this play tried to place this element at the forefront of its delivery by setting Hamlet in some sort of police state with bugs and observation screens to highlight the way in which Claudius abused power and created some sort of Big Brother system of government. Hamlet has no peace or privacy and is constantly observed, watched and monitored. Whilst this is one extreme reading, it does help remind us that really, this play consists of many different, smaller plays with characters playing various parts and roles to try and trick Hamlet. After such an elaborate performance, no wonder Hamlet greets his death looking forward to silence: "The rest is silence." Trying to work out who is being real and who is being sincere and which person is betraying him must have been a nightmare for him. Thus we can see The Mousetrap as representing a motif that runs throughout the play.