As far as the first Blackadder series goes, there never was a Richard IV, after the Battle of Bosworth Field, Henry Tudor claims the crown as Henry VII and the Tudor Dynasty begins. The first series is essentially an alternate history in which Henry never takes the throne, which is instead claimed by Richard Duke of York as the new king Richard IV. The character is entirely fictional, though Richard III did have a nephew Richard who was Duke of York and second son of King Edward IV who disappeared under dubious circumstances during his uncle's rise to power.

In general though, the series tends to play off of traditional stereotypes for laughs (albeit good ones). So as far as any accurate depiction of life in 15th century medieval England goes, we're pretty well out of luck. Edmund would not have been the Duke of Edinburgh, for Scotland was still independent at the time. For that matter, the Scotsman Douglas McAngus is depicted wearing a kilt, which would not come into fashion until the 16th century. Really, there's more inaccuracies than accuracies.

However, there are a couple of interesting points. Although Edmund bungles the thing more often than not, the War of the Roses was an incredibly unstable time in English history, and plots by members and other relations of the royal family to shake up the succession were not out of place or uncommon, Richard III being a prime example. Also, members of the nobility, particularly second or third sons, could enter the clergy and support their family's claims from within the Church, as Edmund does as Archbishop of Canterbury, though I think it would be unlikely for the second son of a king to enter the clergy. By the 15th and 16th centuries, extra princes usually played second-string to the throne, or, in the case that the king was their nephew, acted as a sort of counselor until the king came of age.

And just as an addendum, Edmund's companion, Lord Percy Percy is (loosely) based off the Percy family, who were the Earls of Northumberland during the War of the Roses and significant power players in everything that went down.

TL;DR The first series is intended as an alternate history and the jokes play off typical stereotypes of late medieval England that are based more on popular perception than reality, but like any stereotype, there are some tidbits or kernels of truth that sneak in, buried underneath everything, that help to frame the episodes.