How can I analyze a passage from Hamlet, exploring Shakespeare's use of language and dramatic effects?

I can give you a couple of focal points to consider about Shakespearean language and the tragedy Hamlet overall to use on the passage assigned by your teacher. 

MeterShakespeare is particularly well-known for his use of Iambic Pentameter in his plays. This meter has several uses when it comes to analyzing his work. Noble characters tend to speak in iambic pentameter, whereas common or comic characters speak without form, which contributes to characterization. Syntax...

I can give you a couple of focal points to consider about Shakespearean language and the tragedy Hamlet overall to use on the passage assigned by your teacher. 

Meter
Shakespeare is particularly well-known for his use of Iambic Pentameter in his plays. This meter has several uses when it comes to analyzing his work. Noble characters tend to speak in iambic pentameter, whereas common or comic characters speak without form, which contributes to characterization. Syntax has to be reorganized to accommodate the meter, and Shakespeare tends to use this to advantage in order to emphasize certain elements of a statement, such as a theme or conflict. 

Figurative Language
Shakespeare enjoys integrating analogy, symbolism, allusion, and imagery into his writing. Some or all of these might be present in your passage. Each of these has its own literal meaning but then a subtler, inferred meaning that the audience can discover, which adds to the development of the plot, characters, conflicts, and themes.