There are countless possible examples, presented in detail in The Mysterious William Shakespeare, by Charlton Ogburn, and "Shakespeare" By Another Name, by Mark Anderson. Here I want to focus on just one instance, the remarkably close correspondences between William Cecil (Lord Burghley), Oxford's father-in-law, and Polonius, Hamlet's prospective father-in-law.
Burghley was the closest adviser to the English queen. Polonius is the closest adviser to the Danish king and queen. Burghley was a spymaster with a network of informants; he was known to intercept private correspondence and to infiltrate private households with paid operatives. Polonius's main activity in Hamlet is spying; he intercepts Hamlet's letters to Ophelia, and repeatedly eavesdrops on Hamlet.
One of Burghley's hirelings was stabbed by Oxford in a scuffle. Polonius himself is stabbed to death by Hamlet.
Burghley, worried by reports of his son Thomas's wild living in Paris, arranged for Thomas's friends to spy on him and send back reports. Polonius, worried that his son Laertes will embarrass himself in Paris, arranges for Laertes's friend Reynaldo to spy on him and sent back reports.
Burghley was known for his tedious prolixity; so is Polonius. ("More matter, with less art," the queen tells him.) Burghley wrote sententious maxims for the benefit of his son Robert. Polonius delivers sententious maxims for the benefit of his son Laertes. (Burghley: "Neither borrow of a neighbour or of a friend, but of a stranger ..." Polonius: "Neither a borrower nor a lender be ...")
Burghley was the father of Anne Cecil, whose marriage to Oxford was brought to near ruin by Oxford's antagonism and distrust. Polonius is the father of Ophelia, whose romantic relationship with Hamlet is brought to ruin by Hamlet's antagonism and distrust. Oxford believed Anne had been unfaithful to him, and separated from her for years. Hamlet unaccountably implies that Ophelia has been unfaithful to him, impugning her as a whore, and refuses to see her again. ("Get thee to a nunnery" can be read as Elizabethan slang for "You belong in a whorehouse.")
Burghley was widely known and resented for having pushed through a law requiring the British populace to eat more fish; the intent was to stimulate the fishing trade. Hamlet, either mad or feigning madness, identifies Polonius as "a fishmonger."
Burghley liked to recall that his date of birth coincided with the Diet of Worms, a convocation at which Emperor Charles V presided. Hamlet, jesting cruelly after killing Polonius, says, "A certain convocation of politic worms are e'en at him. Your worm is your only emperor for diet."
Burghley's motto was Cor unum, via una. In the First Quarto of Hamlet, the character later known as Polonius is called Corambis. Cor unum, via una means "one heart, one way." Corambis means "having two hearts." Most likely the name was changed because "Corambis" was just too obvious, at least to those in the know.
The correspondences are sufficiently clearcut that many "orthodox" Shakespearean scholars have accepted the Burghley-Polonius connection. These include such notables as John Dover Wilson and A.L. Rowse.
The Stratford player would have had no plausible motive to burlesque Lord Burghley. Nor is it likely he would have dared. Burghley was the most powerful figure in in the realm, other than the queen herself; he was widely feared, and, at a whim, could have had any commoner tossed into jail or subjected to the tender mercies of the torturers. Other writers had been jailed and tortured for less.
Oxford, by contrast, did have a motive. His relationship with his father-in-law was contentious. The two men, opposites in all respects, despised each other. As the highest ranking member of the aristocracy, Oxford could afford to beard the lion. Burghley couldn't touch him.
Of course, many additional parallels between the events in Hamlet and the events in Oxford's life can be drawn. Oxford was captured and later released by pirates, just as Hamlet is. Oxford lost a vast sum of money funding expeditions to discover a Northwest passage; Hamlet says wryly, "I am but mad north-north-west." One of Oxford's closest friends and confidants was his cousin, Horace de Vere, also known as Horatio; Hamlet's closest friend and confidant is named Horatio. Oxford's mother remarried soon after his father's death; Hamlet resents his mother's hasty marriage to Claudius soon after his father's death. And so on. For a discussion of the numerous parallels in Hamlet, see Chapter 16 of "Shakespeare" Identified, by J. Thomas Looney, online here.
Does all of this prove that Oxford wrote Hamlet and the other plays and poems? Well, no. All the circumstantial evidence in the world cannot establish the authorship of the Shakespearean canon with absolute certainty; but as more and more of it comes to light, the explanation that the parallels are merely coincidental becomes, I think, progressively less persuasive.
At any rate, I concur with John Michell, who, in his excellent overview Who Wrote Shakespeare?, writes:
We shall be abused and laughed at for our interest, for even thinking there is any doubt about who wrote Shakespeare... All that we can say in return is that we enjoy the subject, find mystery in it and are introduced by it to the finest literature and some of the greatest, as well as the crankiest minds of our age and culture. It is a harmless, stimulating and instructive subject to dwell upon, which is more than can be said for many other types of obsession. [p. 16]
For an in-depth discussion of the Burghley-Polonius issue, see Mark Alexander's comprehensive four-part series "Polonius as Lord Burghley," beginning here.
Various relevant source texts are helpfully provided here.