In Act I, Scene 4 of "Hamlet," the minor character Marcellus observes that "there is something rotten in the state of Denmark." And he is correct: the "rottenness" in Denmark extends from the monarchy on down and thus corrupts the entire country, as medieval political/theological theory held that monarch and monarchy were indissolubly linked. Hamlet's uncle became king by murdering Hamlet's father; Hamlet's mother is now married to her murdering brother-in-law, an incestuous marriage in the eyes of the Church – and of her son Hamlet.
In this episode, the allusion incorporates the customary inclusion of reference[s] to "red" as well as indicating that appearance does not match reality at the Redmond high school where a beloved English teacher has been murdered. For a final homage to "Hamlet," Jane gets to use Shakespeare's device – a play within a play, designed to "catch the conscience of the king" – to trap his own murderer AND gets to savor his own school recollections and superb memory by quoting numerous rich and evocative lines of dialogue from "Hamlet" as the students present it.