Rice was born in Amersham, Buckinghamshire, England, and he was educated at St Albans School and Lancing College. He is best known for his collaborations with Andrew Lloyd Webber (having helped create many hit musicals for West End and Broadway theatres), and his work for The Walt Disney Company with Sir Elton John. Rice was made a Disney Legend by the company in 1998.

His lyrics have been widely recognised for their innovation and clarity of expression, varied in style from the pseudo-historical Evita to the simple children's lyrics of The Lion King. He is generally regarded as one of the few contemporary masters of lyric-writing, comparable to other lyrical genuises such as Stephen Sondheim.

He was a co-founder and editor of the Guinness Book of British Hit Singles from 1977 to 1996. He has also been a frequent guest panelist for many years on the radio panel games Just a Minute and Trivia Test Match. Rice often jokes that he is most recognised in America for his appearance in the film About A Boy. The film includes several clips from a (real) edition of the game show Countdown on which he was the guest adjudicator. His other interests include cricket (he was President of the MCC in 2002) and maths. He wrote the foreword to the book Why do buses come in threes?

Rice was famously interviewed by Dom Joly in the cult British sketch show Trigger Happy TV. Whilst discussing Rice's autobiography somewhere near Kensington Church Street in London, Dom feigns a severe migraine, prompting Rice to ask "are you OK? - you're not about to die are you?" Dom attempts to continue the interview but after some moments he wanders off declaring, "I'm sorry I can't do this anymore, it's not my scene" - leaving Rice to ponder, "is that normal?"