In the games,
Ryuki is a famous guitarist. In Sun, Moon, Ultra Sun, and Ultra Moon, he can be fought during a Champion title match in the post game, where he is there to spread his rock fame. In the Ultra Sun and Ultra Moon, he can be also fought hanging around the Kantonian Gym in Maile City.
In the anime, he actually is the owner of the Kantonian Gym in Malie City on Ula'ula Island in the Alola Region. He ignores the aesthetics of the rest of the Gym and Tour Guides and does his own thing.
It is possible that Ryuki is based off of Paul Stanley from the band KISS.