Levinson was born in Philadelphia, Pennsylvania and attended the University of Pennsylvania, where he earned a Bachelor's Degree in Economics in 1956. He served in the United States Army from 1957 to 1958 and married actress Rosanna Huffman in 1969. Levinson had a Jewish heritage.
William Link and Richard Levinson began a 43-year-long friendship in 1946, on their first day of junior high school. Both were avid Ellery Queen fans from boyhood and enjoyed mental puzzles and challenges, a characteristic that would spill over into their work.
Beginning with radio scripts, the team wrote plays and then prime-time TV scripts. They went on to co-create and sometimes produce the detective television series Columbo, Mannix, Ellery Queen, Murder, She Wrote (with Peter S. Fischer) and Scene of the Crime, as well as made-for-TV movies The Gun, My Sweet Charlie, That Certain Summer, The Judge and Jake Wyler, The Execution of Private Slovik, Charlie Cobb: A Nice Night for a Hanging, Rehearsal for Murder, and the short-lived TV series Blacke's Magic. The team were proud of creating "intelligent" rather than violent programs.
The partners also collaborated on two feature films, The Hindenburg (1975) and Rollercoaster (1977), and the Broadway show Merlin, featuring the magician Doug Henning.
The team occasionally used the pseudonym Ted Leighton, most notably on the telefilm Ellery Queen: Don't Look Behind You, where their work was substantially re-written by other hands, and Columbo when they came up with stories to be scripted by their collaborators. They used the name as early as 1959 for short stories published in Alfred Hitchcock's Mystery Magazine when the magazine already contained stories appearing as by Levinson and Link. They also used the name for their contribution to the script for Steve McQueen's final movie, The Hunter. Leighton was Levinson's middle name.
In 1979, Levinson and Link received a Special Edgar Award from the Mystery Writers of America for their work on Ellery Queen and Columbo. During the 1980s, they were three-time winners of the Edgar for Best TV Feature or MiniSeries Teleplay, and in 1989 they were given the MWA's Ellery Queen Award, which honors outstanding mystery-writing teams. In November 1995 they were jointly elected to the Television Academy Hall of Fame.
Levinson died of a heart attack in Los Angeles on March 12, 1987 at the age of 52. He was survived by his wife, actress Rosanna Huffman, whom he married in 1969, their daughter, Christine, as well as his father, William. The first Murder, She Wrote spin-off novel, Gin And Daggers, is dedicated to his memory.
In tribute to Levinson, Link wrote the script for the 1991 TV film The Boys, starring James Woods and John Lithgow.