I started in Wikipedia, that mentions that the song was about a real person, Joseph Clark, a mountaineer that was born in 1939 and who was murdered in 1888. I was unable to find any historical record, mostly because the sources on Wikipedia gave me a 404 error.
Click here for a recording made in 1966 collected by Alan Jabbour of Henry Reed.
The Great Plains Dulcimer Alliance has a good article about the man behind the myth, as they were able to (theoretically) access the physical media Wikipeda was referencing.
Old Joe Clark historical marker says "This marker recognizes the Mountain Ballad of Old Joe Clark, who was born in 1839 and murdered in 1885. He ran a still in the moonshining days of the 1870's. The ballad has approximately 90 stanzas and was sung during World War I, then later by soldiers from Eastern Kentucky."
Here is the article about Lisa Clark's research who did genealogical research on Joseph Clark - perhaps she is a descendant? - the article states the following:
According to Lisa Clark's research, Joseph Clark was born in Clay County, Kentucky on September 18, 1839. He was raised on the family farm at Sextons Creek, and married Elizabeth (Betty) Sandlin on January 12, 1857, when he was 17 and she was 15.
When the Civil War began, Joe was one of the first to enlist, even though he was married and had three children. He was 22 years old, stood 5 feet 8 inches, had a fair complexion, with light hair and blue eyes. He became ill during the winter months and was given a Disability Discharge in 1862.
Joe returned to Clay County and resumed farming. He bought 700 acres of land from his father in 1868, and lived in the log house on Sextons Creek that had been built by the Clark pioneers.
Joe began earning a reputation in the local area, and Betty left him around 1864. He lived with several different women and had more children which he raised.
There was a popular break-down tune at the time that did not have lyrics, so some of Joe's friends started making up rhymes to be sung with the tune. From this originated the ballad of "OLD JOE CLARK." Joe is said to have liked the song until some of the more fun loving souls started making up rhymes that were not very complimentary.
He operated a country store near his house and also ran a moonshine still, under license from the state. The still was located in the bottom near his house, and Joe had orchards from which to gather the fruit for brandy and other drinks. He would load an ox cart with whiskey and take it to the round bottoms as well as selling it from his store. Joe had a Spencer rifle which he carried across his lap when riding and is said to have used it to shoot the arm off one of his neighbors when they got into a fight. One story has it that Joe also lost an arm, but J.B. Weaver, who married Joe's great-granddaughter, claimed Joe lost some use of his left arm after he had a fight with the father of John Lucas, who slashed him across the collarbone with a knife.
There are several stories surrounding his death. J.B. Weaver gave this account, as told to him by Joe's son. Joe was living with a woman named Chris Leger and they split up. He then began living with a McKenney woman in his store, renting his house to Chris and her new friend, the brother of Old Jim Howard. Leger and Howard then devised a plan whereby they would kill Joe and she would claim he had left the farm to her. Howard shot and killed Joe on April 22, 1886, near the back porch of the store. Howard then fled to Beattyville, where a few days later while crossing a bridge, he was stabbed to death by two men from Clay County.
Joe is buried in the Clark Cemetery on a hill overlooking the farm at Sextons Creek
Collected in the Publications of the Texas Folk-lore Society, Issues 1-2, published in 1916.
I found a song called "Old Joe Clog" collected in 1917 published in Folk Songs of the South, 1925. The prosody fits the modern melody, but the notation was not recorded.
Here is an unfamiliar melody from the Journal of American Folklore, Vol. 42, published in 1929. The description describes this song being performed in a variety ways with a variety of different figures. The article states the following:
"Although the old-time dancers are unanimous in contending that "Old Joe Clark" is a regular party-game, played with "figures" as fixed and definite as "Weevily Wheat" or "Old Dan Tucker," my own observations do not support this contention. The game as I have seen it played is a wild medley of elements from other games, and the fact that the same tune is played by fiddlers at the "square" dances still further complicates matters. Occasionally some gifted player, with a shrill whoop to attract attention to himself, breaks into a sort of jig or breakdown, while the chorus is sung loudly by the other dancers who gather about him. And Ihave attended two parties in which all figures were dispensed with, and the players simply "paired off" and danced about in couples - a crude imitation of the modern "round" dances introduced by the tourists,except that there was no music save the singing of the players. The following verses were written down for me by Mrs. Emma Chambliss, who heard them sung by an old woman in McDonald County, Missouri." - THE OZARK PLAY-PARTY, BY VANCE RANDOLPH
The Traditional Tune archive cites "Pioneer Western Folk Tunes" by Viola Ruth (1928) as a source for the melody, but a digital copy does not exist. Here is the Traditional Tune Archive's collection of versions of Old Joe Clark.