Which character says "Something is rotten in the state of Denmark?"
This line spoken by Marcellus (and not Hamlet as is commonly believed) is one of the most recognizable lines in all of Shakespeare's works. It appears in 1.5.90.
What is the play within the play in Hamlet?
In 2.2 Hamlet arranges for the players to present The Murder of Gonzago before King Claudius. The drama shows how Hamlet's father was murdered by Claudius, and Hamlet hopes Claudius' reaction will reveal his guilt. Hamlet also refers to The Murder of Gonzago as The Mousetrap in 3.2.235.
Where was Hamlet studying before he returned to Denmark?
Hamlet was away studying at Wittenberg, the renowned university located near Berlin and founded in 1502.
In Hamlet's third soliloquy, what is the meaning of the word "slings?"
To be, or not to be, that is the question:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles,
And by opposing, end them.
(3.1.64-68)
Some argue that "slings" is a misprint of the intended word, "stings". "The stings of fortune" was a common saying in the Renaissance. However, in the context of the soliloquy, "slings" likely means "sling-shot" or "missile". This seems in keeping with the reference to "arrows" - both can do great harm. For a complete analysis of this soliloquy please visit my feature exploring Hamlets great speeches.
In Hamlet's third soliloquy, is it "disprized", or "despised love"? I've seen and heard both.
The Folio edition of the play has "dispriz'd" (under-valued) while Q2 has despis'd. Most critics use Q2.
In Hamlet's third soliloquy, what does "the proud man's contumely" mean?
"Contumely" means scorn or insolent language. It comes from the Middle English word, "contumelie." The pronunciation is "kän - tyüm - le." For an analysis of Hamlet's soliloquies, please see my article located here.
How does Queen Gertrude die?
King Claudius pours wine to toast Hamlet's success and tries to persuade Hamlet to drink the poisoned brew. Hamlet refuses the wine, placing the goblet on the table beside the Queen. Gertrude is thirsty and, despite the King's plea, drinks from the cup. Dying, Gertrude exclaims that she has been poisoned.

