There's only one character from the Harry Potter books who actually existed in real life
As great as it would be if "Harry Potter" was real, J.K. Rowling made sure the series was distinct from real life. There aren't any specific, real-life people who show up in the books.
There's just one exception: Nicolas Flamel.
In "Harry Potter and the Sorcerer's Stone," Flamel is the magician who created the Philosopher's Stone. The stone has several magical properties: it can turn ordinary metal into gold and it produces the Elixir of Life, which makes the drinker immortal. Naturally, Voldemort was after it.
In real life, Flamel was a French scholar and bookseller who lived in the 14th and early 15th centuries. After his death, he got a reputation as an alchemist who discovered the Philosopher's Stone and secretly achieved immortality. Several pseudepigraphical texts about alchemy were attributed to him, so he gained fame as an important figure in alchemy circles.
J.K. Rowling noted on Pottermore that she read about Flamel in her 20s, and incorporated a version of his story in the Harry Potter series. She dreamt about him while writing her first book:
Rowling's version of Flamel graduated from Beauxbatons, a wizarding school in southwest Europe where he also met his wife. The fictional Flamel was rumored to be a major patron of the school later in life, which syncs up with his real-life reputation as a wealthy businessman and philanthropist.
When Harry was in his first year at Hogwarts, in 1991, fictional Flamel was 665 years old. This would also fit with his real-life age; he was born around 1330.
To be sure, Rowling writes about interactions between the magical and muggle worlds in her books. The British Prime Minister chats with the Minister of Magic on occasion, and wizards live among muggles. But for the most part, details are purposefully scarce, and major historical events in the muggle world are only alluded to. Flamel is the exception. He isn't only given a name, but he's a proper character in the series.