He's the owner of the four turtles who fell into the sewers and mutated to become the Teenage Mutant Ninja Turtles. Chet did not see his turtles until fifteen years ago after the incident with the Mutagen Container when he was working in the sewer sanitation.
Chet has a cameo in the first episode of the 1987 animated series but is not mentioned by name. He appears again in season 5, episode 8.
In the animated series from 2003 Chet has some cameo appearances as a child during the turtles flashback scenes.