The claim is FALSE

The story of → Edward Mordake first spread in 1895 through a newspaper article by a writer named Charles Lotin Hildreth. In the same article, there was also the story of a woman with a fish tail, a man with crab legs and a spider with a human face. 
→ In 1896, Edward Mordake's story was published in the book "Anomalies and Curiosities of Medicine" by two doctors and became widespread.
→ Edward Mordake's story contains medical contradictions. Although the original story says that the face on the back of Mordake's neck belongs to a woman, this is not medically possible. Craniopagus parasiticus, a medical condition similar to the one in the story, is a form of parasitic twinning. Parasitic twinning results in same-sex twins. 

Edward Mordake is a Fictional Character

In the alleged post, the story of a man who was born with two faces and later committed suicide was shared as a chain of posts. According to the story, a man named Edward Mordake was born with a second face on his neck. When he was laughing, the face behind him was crying, and when he was crying, he was laughing. The claim was shared by many users in different languages from time to time. The Turkish text of the claim is as follows;

"The interesting story of Edward Mordake, who was born with two faces and committed suicide!"What made the face on the back of Edward Mordake, one of the most interesting people in the world, more frightening was that he cried when he laughed and laughed when he cried.Edward Mordake, who was described by some as noble and by others as an ordinary peasant, is said to have lived in England in the 19th century. What distinguished him from other people was that he was "two-faced". This hypocrisy was not a characteristic problem. Edward really had two faces, one on the front and one on the back, and the one on the back of his head was the same as his front face, only smaller. As a child, Edward didn't realize that people were scared of him and that's why his friends wouldn't go near him, but as he grew older, he gradually realized this.What made the face on the back of his neck even more frightening was that Edward cried when he laughed and laughed when he cried. This face, always sullen during the day, hissed like a snake at night, telling Edward frightening things. When Edward went to the doctors for treatment, none of them would examine him because they thought he was cursed. Unable to bear his other side any longer, Edward committed suicide by hanging himself at the age of 23. He left behind a story that would inspire artists and screenwriters. It is known that Tom Waits wrote the song "Poor Edward" for Edward Mordake. He was also featured in American Horror Story, and J.K Rowling, the author of Harry Potter, was inspired by Edward Mordake when she created Professor Quirrell, who keeps Woldemort alive in the back of his head in The Philosopher's Stone. Although there are various rumors about the authenticity of the story, there is a lot of information on this subject in foreign sources."

Although the name in the claim is Edward Mordake, internet research shows that some sources refer to the person in question as Edward Mordrake. Mordake's bizarre story dates back to the 1800s. According to research, Mordake's story was included in the book Anomalies and Curiosities of Medicine, published in 1896 by two doctors, George M. Gould and Walter L. Pyle. 

Is the Story of Two-Faced Edward Mordake a Real Person?

The authors of the book state that they got Mordake's story from "ordinary sources", that is, laymen, and did not reveal the source. This source seems to have different details of the story. According to the book, behind Edward Mordake's head was the face of a girl as ugly as the devil and as beautiful as a dream. This detail is not included in the claims circulating today.

Conclusion

The claim that the story of Edward Mordake, born with two faces, belongs to a real person is false.