After studying cello with Lloyd Smith and Orlando Cole in addition to music theory at the Curtis Institute of Music, Stone went on to earn a degree from Indiana University. In 1980, he moved to California to work as a music editor with such composers as Georges Delerue on Platoon and other films) and Maurice Jarre (on The Witness).
He went on to write music for various feature films and television series, including the Bruce Campbell western Sundown: The Vampire in Retreat, Pumpkinhead, as well as North Shore and the miniseries, "In a Child's Name". Stone worked on John Hughes films including Ferris Bueller's Day Off and Sixteen Candles (both scored by Ira Newborn). Stone also composed the music for the William Shatner series, "Rescue 911". Stone also scored the PBS Documentary "Medal of Honor" along with Mark Watters. He later wrote compositions for various films including, Summers heat, Never on Tuesday, Tripwire, Vietnam Texas, and Victim of love.
Stone has won several Emmy Awards for Outstanding Music Direction and Composition for Animaniacs and Histeria, as well as Outstanding Original Song, shared with lyricist, writer, creator and senior producer Tom Ruegger, for the main titles of Animaniacs and Freakazoid. Stone shared many of his music direction/composing awards with his team of composers, who included Steve Bernstein, Carl Johnson, Julie Bernstein, Gordon Goodwin and Tim Kelly.
According to Animaniacs writer/producer Paul Rugg, crew members fondly referred to Richard as "The Great Stonini," a sort of musical magician whose compositions and orchestrations often raised the quality of the cartoons to unexpected musical and artistic heights.
Stone died of pancreatic cancer in Los Angeles, California at age 47.