Let's examine the recentness of (and public familiarity with) the English History Plays first.

For the sake of perspective, let's say that it's Friday, the 28th of June, 1613 and you just bought admission to see Henry VIII in its second-to-last performance at the Globe Theatre. At some point in the show, a cannon will be fired backstage and the audience will jump and cheer at the awesome special effects. So if you brought a drink, finish it early. During tomorrow's show, that cannon will misfire and burn the legendary "Wooden O" to the ground.

If you were an English native, you knew the story of Henry VIII (or at least the official Tudor/Stuart approved version) pretty well going in. Even if you were an uneducated and illiterate person who could barely afford the ticket, you could probably summarize the most important plot points if asked. The stories of Henry VIII and Shakespeare's other English kings would have been the central part of your country's cultural narrative as you received it.

So how far removed in time would you be from the events you are about to see portrayed on stage?

Henry's coronation happened in 1509 (104 years before the "present" in 1613). It would have felt like reflecting on the events of 1909 from today. It's conceivable that your great grandparents would have been Henry's subjects.

Let's say this imaginary you is a long-time Shakespeare fan & saw several of the English History Plays in their original run. How distant in the past were they? I'll use the following format:

Title of the play, year debuted (~years since actual events) [2015 perspective]

NOTE: Precise dates of some first performances remain debatable.

Richard II, premiered 1595 (~218 years since events) [like looking back to 1797]

Henry IV Parts 1 & 2, 1597 (~198 years) [1817]

Henry V, 1598 (~185 years) [1830]

Henry VI Parts 1, 2, & 3, 1591 (~162 years) [1853]

Richard III, 1592 (~109 years) [1906]

So while the events of the English History Plays had passed out of living memory, they would not be very far removed from cultural memory.

But what about Caesar? The story was widely known and taught, though not as extensively studied as the local history. If this imaginary you was formally educated as a child, you would have learned the tale in school. Shakespeare's Julius Caesar premiered in 1599, 1,643 years after the events depicted in the play.

Plutarch’s Lives of Julius Caesar and Marcus Brutus was translated into english in 1597 by Thomas North. It was a very popular seller and is widely believed (almost certain) to have been used by Shakespeare as a source while writing Caesar. Books were expensive in the late 1500s, but if you were an educated person of privilege you could have purchased a copy and read it before seeing the premiere.

Follow-up question: What kind of people would have seen Shakespeare's plays in the late 16th/early 17th century? I've read claims that they were the "soap operas of the day" but did they actually have a mass audience? And were the plays performed outside London, or even England?

The schools of the time didn't cover English history (nor English grammar, either), and in fact the English theatrical industry was given a big shot in the arm when Elizabeth I came to power and used the theatre for propaganda to engender a sense of Englishness in the people to unify the country by giving them a patriotic, pro-Tudor version of English history, while at the same time luring them away from Catholicism and toward the state church. The theatre had been traditionally used by the Catholic church to entertain and edify the population, but Elizabeth outlawed religious playing and secularized the playing companies and made them answerable to the crown by requiring that they be sponsored by aristocrats. Along with censorship of the press and stage, that ensured that subversive ideas were prevented from being spread amongst the population (which was often circumvented by clever playwrights by writing allegorically). So to answer your question, the average person would not have known English history except by traditions and hearsay, since every area of the country was owned by aristocrats who had allegiance to the crown and who were required to provide soldiers from his area for the wars of the crown. Churches also educated the populace to an extent by providing moral support for the wars and by being the repositories of elaborate tombs of the illustrious inhabitants of the area.

Classical history was most often learned by the study of Latin in both grammar schools and universities. Grammar school attendance and literacy increased greatly during the Early Modern era as the bourgeois class and the government expanded hand-in-hand, but I don't think it could be said that the average person was literate. Only about 20% of the males and 5% of the females of the time could sign their names, which is a rough indicator of literacy. However, many people could read without knowing how to write, but it is doubtful that even half the population could. See David Cressy's Literacy and the Social Order: Reading and Writing in Tudor and Stuart England (1980).