First, the Hamlet parallels. All right, maybe I should clarify. When I said that all the elements of Hamlet come from Shakespeare's sources, I didn't mean that they were all in Saxo Grammaticus and Belleforest, which is where the basic story came and some of the incidentals came from. A more immediate source was Kyd's Spanish Tragedy, which has a ghost, a play-within-a-play used for an ulterior motive at court, a hero who reproaches himself for delay and considers suicide, a woman whose love is opposed by her father and brother, a woman who goes mad and kills herself, and an avenger and his intended victim who have a public reconciliation before an ultimately tragic end. More generally, the elements of Hamlet were fairly common motifs of Elizabethan tragedy which the audience would recognize and expect in their afternoon's entertainment. Inevitably, some of these motifs can be seen as bearing resemblances to real-life people, especially when those people are members of the nobility, but the sheer number of such proposed parallels (some of them quite ingenious but many of them mutually exclusive) tends to make scholars very skeptical in the absence of external evidence, and with good reason.
For the specific case of Hamlet, I mentioned King James and the Earl of Essex, so let me outline the cases for them as Hamlet. James's father, Lord Darnley, had been murdered; his mother had been suspected in the scandal of his death, and she soon married the supposed murderer, Bothwell (who was a heavy drinker, just like Claudius). Mary's meddling chief counselor, Rizzio, was murdered in her presence, and his body was disposed of secretly by means of a stair-case. Sound familiar? James was a melancholy, indecisive prince, interested in learning, a poet, married to a woman (Queen Anne) who he treated shabbily, and a likely successor to the throne of England. How can you not see the parallels? Hamlet is essentially James's life story. If that doesn't get you, take a look at Essex. Rumor had it that the Earl of Leicester had poisoned Essex's father, the first Earl, in order to live in sin with Essex's mother, Lettice Knollys. Essex was married to Frances, daughter of Sir Francis Walsingham, Elizabeth's Secretary of State, head of her secret police (thus Polonius's spying), and rival to Burghley for the title of her chief advisor; the match was opposed by the bride's family (unlike in Oxford's case). Essex was highly educated and addicted to learning, a moody, brilliant, and unstable man who liked to wear black, a notorious procrastinator, sometimes abusive to women (including the Queen), an excellent poet and a patron of players. If you want to consider Polonius a composite of Burghley and Walsingham (very reasonably), then I could add that Essex was an enemy of Burghley. How can you not see the parallels here? Hamlet is essentially Essex's life story; I would submit that the parallels are closer than they are for Oxford (for another discussion of the supposed parallels between Hamlet and Oxford's life, see Irvin Matus's essay The Oxfordian Hamlet).
Ah, but what about the other plays, "E of O" says? Well, there are fistfuls of parallels to Essex there, too, and more than a few to James. In 1591 Essex banqueted with Navarre, Biron, and Longueville, the real-life namesakes of the characters in Loves Labours Lost; Dover Wilson's Cambridge edition of 1 Henry VI persuasively argues that Talbot is modeled on Essex at the siege of Rouen; many commentators have pointed out persuasive parallels between Essex and Bolingbroke and Henry V; Robert Cartwright argued very plausibly in 1863 that Essex is Romeo, Antonio in Merchant of Venice, and Achilles in Troilus and Cressida, among others. Taken together, I find the Essex parallels in Shakespeare considerably more striking than the supposed Oxford parallels; I haven't even gone into the parallels to King James, or Sir Philip Sidney, or others. (By the way, "E" mentions the bed- trick in All's Well and Measure for Measure as though it's something unusual, but this was an extremely common device of Elizabethan theater; see the recent book The Bed-Trick in Elizabethan Drama for many examples.)
One more thing on this topic. "E" drags out the standard Oxfordian argument that Polonius was modeled on Burghley, and how could a commoner like Shakespeare know enough about Burghley to lampoon him, let alone get away with such impudence? Well, we had this argument last year on SHAKSPER, and I don't want to repeat all that, so I'll just say this. I don't know whether Polonius was partly modeled on Burghley; some of the Oxfordian arguments on this point are a mighty stretch, but you can make a respectable case. Even if he was, that is absolutely no reason to say or imply that William Shakespeare could not have written Hamlet. First of all, we have abundant evidence that court gossip was extremely popular at all levels of Elizabethan society, and that Burghley was one of its most popular topics. For example, John Manningham's Diary, written in 1602-3, has several unflattering anecdotes about Burghley, and the man had been dead for four years. (The diary of Manningham, a commoner, is full of court gossip, as are the letters of John Chamberlain, another commoner.) Spenser's Mother Hubbard's Tale, published in 1591, contained a vicious parody of Burghley in its fable of the Fox and the Ape, and we know from external evidence (a letter dated March 19, 1591) that Burghley was widely known to be the target. Thomas Nashe also parodied Burghley in Pierce Pennilesse, and D. Allen Carroll has recently made a strong case that Burghley was attacked in the notorious Greene's Groatsworth of Wit. If these commoners could attack Burghley, why couldn't Shakespeare, who as a member of the Chamberlain's Men often played at Court, where he undoubtedly had access to the latest gossip?