WARNING: The following contains minor spoilers for the Demon Slayer: Kimetsu no Yaiba manga by Koyoharu Gotouge, available in English through Viz Media.
The Hinokami Kagura, aka Sun-Breathing, is one of the biggest mysteries in Demon Slayer. The original, most powerful Breathing Technique was once wielded by a man whose encounter with Muzan Kibutsuji still terrifies the Progenitor of Demons to this day. Tanjiro Kamado is the only (known) living user of the Hinokami Kagura, but unlike the Water-Breathing he learned from former Hashira Sakonji Urokodaki, Tanjiro learned Sun-Breathing from his and Nezuko's late father, Tanjuro Kamado.
So the question now is: who was Tanjuro Kamado? And why was a charcoal-burner living on a mountain a master of the most powerful version of the Breathing Styles used by the Demon Slayer Corps?
Tanjuro Kamado was assuredly not a Demon Slayer, considering the family keeps records that denote the Kamados as charcoal-burners going back several generations. Unlike the Rengoku family who form a dynasty of Demon Slayers, the Kamados did not even know demons existed until the fateful night that Kibutsuji happened upon their home and slaughtered them all. Though Tanjuro was not a Demon Slayer, Muzan is fortunate that he had already passed away when the villain attacked the Kamados, as Tanjuro was a master of the very Sun-Breathing he dreads so much.
Tanjuro passed away from an illness sometime before the events of Chapter/Episode 1 of Demon Slayer. Tanjiro remembers his father as a gentle, frail man who would dance the Hinokami Kagura every year, even as his health began to decline. The Hinokami Kagura and Sun-Breathing are one in the same, yet Tanjuro did not use it to kill demons, but rather as a ritualistic dance and form of prayer to ward off injuries from the fire they worked with as coal-burners. Tanjiro noted that his father could perform the Hinokami dance throughout the entire night, even in the bitter cold and suffering from his illness without growing the least bit weary.
It's unclear how much, if anything, Tanjuro knew about demons or the Demon Slayer Corps, but he did know about Breathing Techniques. When a young Tanjiro questioned how his father could do the Hinokami Kagura all night without getting cold or tired, Tanjuro answered that there is a certain way to breathe to ensure one never grows weary. Yet for a long time, Tanjiro couldn't perform the Hinokami Kagura with Breathing Techniques without exhausting himself after a single move.
Tanjuro's influence on Demon Slayer is through his relationship with his son, as it's unclear how much Nezuko remembers of him. He seemed to be grooming Tanjiro to be his heir in passing down the Hinokami Kagura. In a flashback during the manga's yet-to-be-animated "Infinity Castle" arc, mere weeks before Tanjuro died, he took Tanjiro into the woods to confront a large bear that had eaten several people nearby.
Tanjiro was awed that his father knew the bear was there, even though it hadn't rung the alarm bells they'd strung up. With a hand ax far too small to be of any use against such a large animal, Tanjuro decapitated the bear in the blink of an eye. So, not only could Tanjuro use his mastery of the Hinokami Kagura for his dance, but he was also an excellent warrior despite his poor health.
Tanjuro Kamado bore a mark on his head similar to Tanjiro's own scar, identifying Tanjuro as a chosen wielder of Sun-Breathing. This confirms Shinjuro Rengoku's implication that Sun-Breathing is a thing of destiny. Muzan Kibutsuji is famously anti-Sun-Breathing, so it's either fortunate or by design that he didn't attack the Kamados until after Tanjuro had passed away. Tanjiro's own near-mastery of Sun-Breathing is almost certainly the Demon Slayer Corps' greatest hope for ridding Japan of demons. If Tanjuro had lived long enough to properly train his son in the ways of Sun-Breathing -- or fight using his own mastery of the Style -- Demon Slayer may have been a much shorter series.
