Keller was born in Chicago and attended University of Illinois, where he began writing comedy with his fraternity brother Allan Sherman. He served in the Pacific Theater with the United States Army Signal Corps during World War II.
After the war, Keller came home and married Bernice "Bitsy" Berkowitz. They had two children, Casey and Jamie. In 1951, he borrowed $500 from his father-in-law and moved the family to New York hoping to become an entertainer and comedian. He soon began writing for television.
On Caesar's Hour, starring Sid Caesar, Keller worked with notable writers Mel Brooks, Carl Reiner, Selma Diamond, Larry Gelbart, Mel Tolkin, Michael Stewart and Gary Belkin. In 1956, 1957 and 1958 the show was nominated for Emmy Award for Best Comedy Writing – Variety or Situation Comedy.
Keller also wrote several episodes of The Dick Van Dyke Show and M*A*S*H, including "For Want of a Boot" and "The Chosen People". His notable screenplays include Buona Sera, Mrs. Campbell (written with Melvin Frank and Denis Norden), which would later become the inspiration for the stage and film musical Mamma Mia!
In the early 1980s, as his writing career was winding down, Keller formed the Beverly Hills Unlisted Jazz Band with friends Conrad Janis and George Segal. Their mix of jazz and comedy made them popular enough to play at Carnegie Hall and on The Tonight Show and led to their own PBS special in 1993, "This Joint Is Jumpin'".
In later life, Keller collaborated with his friend Howard Albrecht on Funny Stuff, a newsletter of jokes for radio DJs and public speakers. Keller died at his home in Valencia, California, from complications of Alzheimer's disease.