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.