Discuss how Hamlet's "Now I am alone" soliloquy contributes to the plot, characterization, and atmosphere of the play.Analyze the "Now I am alone"...

Many people consider this soliloquy to be a turning point
in the plot of Hamlet.  Actually, in my Arden text, the line
numbers are 543- 601.

In this conversation with the
audience, Hamlet considers the invented reactions of an actor to the pretend
circumstances of the text he speaks to his own behaviour in reaction to real events in
the true circumstances of his own life.  The upshot of the speech is the birth of
Hamlet's idea to create a pretend re-enactment of his father's murder and have it
performed before Claudius.  Hamlet determines that the witnessing of Claudius' reaction
to this will reveal whether Claudius murdered his father or
not.

The idea to present the play-within-a-play
(The Mousetrap) to Claudius certainly advances the plot because the
audience has no more information about Hamlet Senior's murder than Hamlet, as of yet. 
This gives the audience cause to also be very curious to see what Claudius' reaction
will be.  Did he do it?  This play might very well prove it.  So, the plot and action of
the play picks up steam at the end of this soliloquy.

It
adds to the atmosphere by creating suspense for just the same reason it advances the
plot.  The audience's expectation is heightened, and it is ready to see what will happen
next.  The decision to present the play seems to put Hamlet that much closer to actually
revenging his father's death instead of just talking about
it.

Of course, all of the things mentioned above also add
to the audience's understanding of Hamlet.  But the soliloquy also gives further insight
into the sort of overly self-critical aspects of Hamlet's nature.  He spends the first
part of the soliloquy comparing himself to the actor, and railing against and condemning
himself for being unable to act:

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Yet I,

A dull and
muddy-mettled rascal, peak

Like John a-dreams. .
.

And can say nothing. . .

. .
.Am I coward?

'Swounds, I should take it.  For it cannot
be

But I am pigeon-liver'd and lack
gall

To make oppression bitter, or ere
this

I should ha' fatted all the region
kites

With this slave's
offal.

And so, it is out of
this non-action, this self-condemnation (and condemnation of Claudius) that the idea for
an action is born.  The play-within-a-play is the first (and only) real action against
Claudius that Hamlet takes until the duel at the very end of the play, which makes this
soliloquy a very important turning point in the play.