What are some points to remember about Hamlet when writting a passage analysis on Shakespeare's Hamlet?

These ideas might help:

1.
 Establish the context of the passage.  Try to determine how the passage fits into the
play as a whole.  Does it come at the beginning, middle, end?  Is it a "first"?  For
instance, is it the first time we see Hamlet  and Ophelia together on stage?  Is it
Hamlet's first soliloquy? Is it a "last"?  The last time we see Ophelia on stage?  The
last episode of spying in the play?

2.  Relate the passage
to overriding themes of the play:  family honor, effects of revenge on the avenger,
betrayal, action versus inaction, death, madness.

3.
 Determine what kind of speech is being addressed: soliloquy, ceremonial or public
speech, dialogue.

4.  Consider characters who are on stage
who may not have speaking parts.

5.  Address the structure
of the passage.  Is it in prose (usually indicative of madness or feigning madness)? Is
it in verse form?

6.  How is the passage itself organized?
Does it involve shifts in tone?  Does it move from dialogue to soliloquy?  Contemplation
of inaction to a plan for action? Or vice versa? Is it arranged as  a philosophical
argument in which the pros and cons of a decision are addressed, as in Hamlet's "To be
or not to be" soliloquy?

7.  Look at the figures of speech
in the passage or images or motifs that are prevalent throughout the play:  disease,
rottenness, garden, for example.  Be sure to relate a discussion of these to the ideas
presented in the passage.

8.  Pay attention to sound
devices such as alliteration, assonance, monosyllabic lines that serve to highlight key
words and ideas.

9.  Be sure to address the first and last
lines of the passage.

10.  Consider the passage's impact on
the audience.  What reaction does the audience have upon viewing or reading this scene?
 Does it change our feelings toward the characters or their situation?  Does it create
more pathos? Less?

11.  Lastly, what would be lost if this
passage were cut from a production of the play.

You may
structure your analysis as you see fit, but I would suggest that you begin with a
general introduction stating the context of the passage, its major purposes.  Then
discuss the passage more or less in a linear fashion, perhaps even dividing the passage
into thirds and discussing how each third contributes to the overall purpose of the
piece.  Shakespeare arranged the passage in a logical way, and following his arrangement
to guide your remarks is a sensible way to attack such an
assignment.