What are the key Elizabethan ideas we need to know in order to understand Hamlet?The importance of order/harmony, the Great Chain of Being, the religious beliefs, family structures, etc?
One belief that is necessary for an understanding of Shakespeare's Hamlet is the belief that Gertrude's marriage to Claudius is an act of incest. The marriage, therefore, is not just unusual, and is not just uncomfortable for Hamlet, it is sinful.
Gertrude would have become one with King Hamlet's family when she married into it. Marrying King Hamlet's brother (Claudius) was seen as marrying a member of her own family. Thus, the marriage is incestuous, as Hamlet says in the play.
This helps to explain Hamlet's obsession with the marriage and, particularly, with Gertrude's sex life. Gertrude having sex with Claudius, viewed by Elizabethans, is an incestuous act. She is living in sin, in Hamlet's eyes.
Hamlet has not only lost his father unexpectedly, lost his rightful throne, and experienced his mother's hasty remarriage, but he has also experienced his mother committing incest to satisfy, as he sees it, her sexual desires.
Gertrude's marriage to Claudius is not so hideous to a modern audience. To fully appreciate Hamlet's situation, however, one must understand the Elizabethan idea that the marriage is incestuous.
There is an Elizabethan idea that you don't mention which is quite central to the plot of Hamlet -- revenge, or more specifically, a Revenge Tragedy. Revenge was quite a popular play topic, and each Revenge Tragedy's plot revolved around the protagonist's avenging of a morally unjust murder.
In Hamlet, this concept is followed through. Hamlet does, in the final scene of the play, avenge his father's murder. However the questioning of whether the ghost was actually a messenger bearing a true report or a demon bearing a false one, of whether or not (Hamlet decides not) to kill Claudius while he's praying and send him to heaven, and the questioning of the usefulness of any sort of action, are all new additions to a standard Elizabethan Revenge Tragedy plot structure.
And by, causing Hamlet to question the act of revenge itself (which, in Revenge Tragedies that had come before was performed without question), Shakespeare created a play that has risen above its genre as an incomparable comment on how a human being decides whether or not to act, even when that action is the avenging of a morally unjust murder.