Not being very familiar with Hamlet (I think I read it once a very long time ago, and have seen various film adaptations, but don't recall much), I wasn't aware of the various references and allusions being made to it in Infinite Jest. One of the most interesting that others have pointed out is that the first two words of Hamlet are "Who's there?", and the first two words of Infinite Jest are "I am". I find this interesting because the most overt Hamlet references in the novel revolve around Hal and Himself. i.e. Hal holding up JOI's skull a la Hamlet holding up Yorick's; JOI appearing at the end of the novel as a ghost; JOI's production company being titled 'Poor Yorick Entertainment'; Hal perhaps being a central 'hero of inaction', a la Hamlet and Hal's own essay on the same subject; the title of JOI's 'perfect entertainment'. And the exchange of "Who's there?", and "I am" is a good way to to give a cursory summary of the relationship between JOI and Hal. i.e. JOI trying to reach Hal, and get him to 'speak', or become less solipsistic. Hal declaring who he is in response, clearly (if not audibly). That, as he says a few lines after the opening two words, "I am in here". Whether who he is internally manifests itself outwardly or not.
Also, as to the character relationships that mirror Hamlet, to some degree. I found this overview in the sparknotes for Hamlet:
"When Horatio and the watchmen bring Prince Hamlet, the son of Gertrude and the dead king, to see the ghost, it speaks to him, declaring ominously that it is indeed his father’s spirit, and that he was murdered by none other than Claudius. Ordering Hamlet to seek revenge on the man who usurped his throne and married his wife, the ghost disappears with the dawn.
Prince Hamlet devotes himself to avenging his father’s death, but, because he is contemplative and thoughtful by nature, he delays, entering into a deep melancholy and even apparent madness. "
Clearly there are some heavy plot and character parallels here, that might even be clues to interpreting the events of Infinite Jest. Prince Hamlet = Hal, King Hamlet = Himself, Claudius = C.T. and Avril = Hamlet's wife. However, there is no suggestion anywhere that JOI's death was anything but a suicide. Though there is no suicide note, and JOI's wraith never explicitly admits to 'eliminating his own map'. Further, as outlined in previous blogs, there are a lot of hints and clues that Avril has connections with Quebecois separatists, and had nefarious intentions re: Himself and his work. Recall the end of the novel where Orin is captured by the AFR and placed inside a giant tumbler, and they release roaches, his greatest fear, into the tumbler with him. How did they know his greatest fear? They had sent their female operative to be intimately involved with him and learn these details, and are now exploiting them. Perhaps Avril functioned in the same way to JOI, acting as an embedded operative. JOI was an important political figure for numerous reasons. Recall Hal and Mario's discussion about Avril not being sad about Himself's death and (get this) always having to travel to various locations for meetings of various kinds by herself and now she stays at home, always, since JOI's death. Suggesting that her going to meetings was so that she could relay information about JOI to the separatists groups, who have gotten a hold of her and recruited her. And after JOI was gone she no longer had any reason to go to the clandestine meetings. Further it's possible that, in the same manner Orin's deepest, darkest fears were discovered and exploited by the AFR, so to were JOI's (i.e. the fear of his son being terminally solipsistic), and that it was through a similar method that Avril (perhaps in league with C.T. and other separatists) forcefully, intentionally drove JOI to suicide (recall the Wild turkey 'gift' from Avril at the scene of his suicide). Which would complete the parallel between Hamlet. i.e. C.T. and Avril 'killing' JOI for political purposes, and immediately getting together afterward, while the 'mad' son Hal can not, or will not, has some mental block against, connecting the dots. Then JOI's ghost tries to help Hal connect the dots, but, by the 'end' of the novel, he still hasn't (though he probably has by the time he's digging up his father's head).
Speaking of Avril and chapter 1, where is she?