In Claudius' first speech to Hamlet about his grief over his dead father:
Why should we in our peevish opposition Take it to heart? Fie! 'tis a fault to heaven, A fault against the dead, a fault to nature, To reason most absurd: whose common theme Is death of fathers, and who still hath cried, From the first corpse till he that died to-day, 'This must be so.' We pray you, throw to earth This unprevailing woe, and think of us As of a father...
Who was that first corpse he talked about there? Abel. A man who was killed by his own brother!
When Claudius prays for forgiveness in the same scene, he says something to the effect of his crime being subject to the "most primal curse." But, of course, Claudius comes out of a very Christian background. What anecdote in Christian teachings talks about a brother killing a brother?
If you subscribe to the idea that Gertrude was a knowing participant in her late husband's murder, the causes of death in the last scene make perfect sense: The two gallant sons avenging a murdered father's death is killed by the sword in honest combat while the villainous conspirators are killed by poisoned drink, often called a coward's weapon.
Hamlet asks Gertrude, "How like you this play, Mother?" Depending on inflection, he could be asking her whether she was enjoying the play, or if she had realized the play resembled her circumstances.
Claudius' line, "My offense is rank; it stinks to heaven" is layered and complex, a fusion of the tragic and the comic. Claudius' offense is 'rank' both in the sense of an unearned position (his rank as king) and in the sense that his offense is figuratively smelly and repulsive. It "stinks to heaven" both in the sense of a rising stench, and also in the sense of being repulsive to the heavens, an affront to the sacred.
Hamlet gets in a Stealth Insult to Claudius on two fronts with these lines:
On the surface it's him just barely stopping short of calling Claudius an ass, as lampshaded by Horatio's dry, "You might have rhymed." However, if King Hamlet is Jove/Zeus, then Queen Gertrude would be Juno/Hera. The peacock is one of Hera's sacred animals, transformed from her watchman Argus. Hamlet might be deriding Claudius as supremely unworthy of the throne, as Argus (and indeed a mere peacock) is to Zeus.
After they finish depicting an all-too-accurate version of the murder of Hamlet's father, what happens to the Players? Claudius likely has them interrogated and imprisoned to determine the source of their information and to prevent it from spreading.
It astounds many viewers how vicious Hamlet became toward Claudius. Old Hamlet only told Young Hamlet to avenge his death. Nothing about making sure he's punished, just kill him. Right after the players leave, Hamlet is presented with a perfect chance to kill his uncle, but he doesn't. Why? Because his uncle is praying, and he doesn't want him to go to heaven. He doesn't just want to fulfill his father's wishes, he wants to send Claudius to Hell.
If Rosencrantz and Guildenstern were aware that the letter they were to deliver contained an order for Hamlet's death, they had no reason to deliver it. If they were not aware, then they would have delivered it.