The short answer is that generally they don't.  However, the original Oxfordian, J. Thomas Looney, perceived certain parallels with the life of the 17th Earl of Oxford.  Here are two prominent examples:

  • The male lead in All's Well That Ends Well is Bertram, Count of Rousillon.  As portrayed in the play, the fatherless young count becomes a royal ward; he resists a marriage which has been arranged for him; he travels abroad, leaving behind a loving and distressed young lady; he yearns to serve in military campaigns, but is commanded by his monarch to remain at Court.  Subsequently he goes away without permission.  On two occasions Shakespeare seems to make a slip by referring to him as an earl (III, v, 12 and 19).

  • In Hamlet the most plausible potential caricature is the king's advisor, Lord Polonius, who in some respects resembles Queen Elizabeth's chief advisor, Lord Burghley.  In Act I, Scene iii, Polonius gives life-advice to his departing son, Laertes, in the form of what he terms precepts.  Around 1584, Burghley had also provided his son, Robert, with life-advice in the form of a letter of ten precepts.

Oxfordians (and some others) point out that all of the above-described circumstances of Bertram match the personal history of Edward de Vere - too close a fit to be dismissed as coincidence, in their opinion.  And, they say, only someone close to Burghley - like his one-time ward and son-in-law, Oxford - could know of the precepts he gave to his son (though these are hardly original in their thrust and bear little resemblance to those enunciated by Polonius, as can be seen at http://www.princehamlet.com/burghley.html).

However, there is another person who shares such history and connections. Shakespeare's patron, the third Earl of Southampton, was also a ward of Burghley, as was his great friend, Robert Devereux, Earl of Essex. Strong echoes of Essex and Southampton are perceivable in the characters of Hamlet and his friend Horatio, respectively. And Southampton is at least as good a match as Oxford for Bertram, since, unlike Oxford, both Southampton and Bertram won battle honours in the rank of "General of the Horse".

Interestingly, Bertram's character is far from admirable.  He is depicted as lascivious, vainglorious, rash and unworthy; a snob and a liar, who does not keep his promises or pay his debts.  We cannot rule this out as a caricature of Oxford (though it is unlikely he would so depict himself, if he were the author).  On the other hand, the unflattering portrayal is well explained by the later history of Shakespeare and Southampton (see link below).

These two examples illustrate a circumstance generally given insufficient weight: the remarkable correspondence between the lives of Edward de Vere and Henry Wriothesley, third Earl of Southampton.  Each was an earl and courtier, was an only son who at a young age lost his father, became a ward of Burghley, had problems with a step-father, studied at Cambridge and Gray's Inn, was initially refused leave by the Queen for military service and foreign travel, travelled nevertheless, was summoned back by the Queen, eventually served in both army and navy, could be quarrelsome and petulant, had high profile ructions on or by the tennis court, plotted against and was imprisoned by the Queen, had difficulties with marriage and finances, displayed bisexual behaviour, impregnated a Queen's maid of honour, raised three daughters, had connections with France and Italy, was forbidden by the Queen to duel, fell out with Burghley, officiated at the coronation of King James, patronised literature and theatre and had personal connections with actors.

The truth is that any perception of an Oxford-Shakespeare parallel is most likely an illusion, caused by genuine correspondences in the lives of Oxford and Southampton.  The perceived Shakespearean parallels, if not reasonably explainable as coincidences, can be construed far more plausibly as a by-product of Shakespeare's intercourse over many years with the Earl of Southampton - an intimacy fleshed out in Biography in the Sonnets.