What is Hamlet's purpose in the play?Consider tensions, complications.

The first step when analyzing any character's purpose in a
play, is to have a clear grasp of the play's structural base, so that the character
might be seen as filling a necessary role within that structure.  Hamlet
is a tragedy, and, as such, requires a tragic hero who suffers from a flaw
that is the instrument of his own demise.  Most tragic heroes recognize their flaw as
the reason for their own demise, rather than blaming it on outside
events.

Hamlet's purpose is to serve as the play's tragic
hero, and his flaw is his own hesitation, his inclination to analyze and consider to
excess, rather than react and take action to kill Claudius immediately.  This flaw sets
in motion the complications, the events that come out of
Hamlet's inability to act swiftly and directly against Claudius, most notably, the death
of Polonius.  From this death comes the madness and death of Ophelia; the anger and
action-oriented reaction of Laertes (a foil to the inaction of Hamlet at the murder of
his own father); and the banishment of Hamlet and deaths of Rosencrantz and
Guildenstern; not to mention the final bloodbath that ensues from the duel instigated by
Laertes and Claudius -- all in response to Polonius'
death.

Of course, Hamlet's inaction also assists in the
tension and suspense created in the play.  When will he
finally rise up and strike?  The audience is able to wonder this throughout the
play.

So, Hamlet's purpose is to serve as the tragic hero
of the play, his inaction (and the act of murdering Polonius) creating a major portion
of the play's complications and tension.  For more on Hamlet's purpose in the play,
please consult the links below.