Which, of Hamlet's seven soliloquies is most important to the development of both the Hamlet's character and the plot?
Hamlet's most important soliloquy in terms of character and plot development is in Act 3.1.64-99.
In these verses lie the most important philosophical and moral quandaries with which Hamlet (and indeed, humanity) struggles: what is the point of living? Of being? Is it better to endure the hard times, knowing that there is no end to hard times or end life yourself? Why do we struggle with conscience (and consciousness?)
As Hamlet discovers, there will be no easy, or even satisfactory, answers to any of his questions. He determines his character, and the unfolding of the plot, by wrestling with each one of these dilemmas.
Hamlet's internal turmoil has resonated with playgoers and readers for centuries. Some of the most familiar quotations in all of Shakespeare are found here: "To be or not to be -- that is the question"; "To die, to sleep --/To sleep perchance to dream"; " when we have shuffled off this immortal coil";"conscience doth make cowards of us all."
... To die: to sleep;
No more; and by a sleep to say we end
The heart-ache and the thousand natural shocks
That flesh is heir to, 'tis a consummation
Devoutly to be wish'd. To die, to sleep;
To sleep: perchance to dream: ay, there's the rub;
For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause: ...