In terms of characterization, this soliloquy shows us the continuation of Hamlet's melancholy and his self-depracating attitude about his lack of action to this point in the story. He opens the speech with a metaphor: "O, what a rogue and peasant slave am I!" He realizes that he is slave to his intellect and that he is not able to tap into the raw emotion of his situation and just kill Claudius. He later says that he is "a dull and muddy-mettled rascal, peak(ing) like John-a-dreams, unpregnant of my cause." Here is calls himself a day-dreamer who is caught up in thoughts and not action. He says he is unpregnant, meaning he is not full of life or action for his cause which is to avenge his father's murder. He is fully aware that he is not acting!
He goes on to ask if himself if he is a coward or a villian. He concludes that he is pigeon-livered and lacks gall -- both suggesting that he is, in fact, a coward. He concludes by calling himself an ass -- it meant then the same as it means today -- he is a jerk who can't summon up the verve or energy to do what he knows he needs to do!
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:
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.