Rothschild was born in Paris, the youngest child of Victor Rothschild, 3rd Baron Rothschild and his second wife, Teresa Georgina Rothschild (née Mayor). On his father's side, he descended from the Jewish Rothschild family. His half-brother was Jacob Rothschild, 4th Baron Rothschild. His maternal grandfather, Robert John Grote Mayor, was the brother of English novelist F. M. Mayor and a greatnephew of philosopher and clergyman John Grote. His maternal grandmother, Katherine Beatrice Meinertzhagen, of English and German descent, was the sister of soldier Richard Meinertzhagen and the niece of author Beatrice Webb.
He attended King's College School, Cambridge and was then educated at The Leys School. He graduated in 1976 from City University (London), where he read economics, history and archaeology. His initial vocation was in journalism, as a circulation manager of the now defunct literary magazine, The New Review.
In 1987, Rothschild joined the family banking firm, N M Rothschild & Sons. By 1990 he was made chief executive of one of the Rothschild's weaker assets, Rothschild Asset Management. In 1993, he became Chairman of the firm, helping to steer it onto a more stable path.
One of his favourite pastimes was farming. Rothschild had inherited an arable farm, Rushbrooke, near Bury St. Edmunds where he spent many hours at work. He was also a keen racer of cars, and was known as one of the best amateur racers in the country. His passion started when he acquired a scrambler motorcycle, at age thirteen. In later life, he got into motor racing, and was the winner of the Peter Collins Memorial Trophy Race at Silverstone in 1996. Among his car collection was an AC Cobra Daytona sports car, a 1957 Maserati 250F Formula I, and a 1958 BRM P25 Formula I. He was known as one of the best classic car racers of his generation.
Rothschild was also an amateur pilot, and kept a Stampe at his home in Suffolk.
Rothschild was a trustee of Yad Hanadiv, the Rothschild foundation which has supported many projects in Israel, including the building of the Supreme Court and Knesset in Jerusalem.
In 1981, he married Anita Patience Guinness, daughter of the merchant banker James Edward Alexander Rundell Guinness and Pauline Vivien Mander. They had three children:Kate Emma Rothschild (born 1982) who married Ben Goldsmith, a son of the late billionaire Sir James Goldsmith and Lady Annabel Goldsmith), in 2003 at St. Mary's Church in Bury St. Edmunds. They have three children: Iris Annabel Goldsmith (b. 2004), Frank James Amschel Goldsmith (b. 2005) and Isaac Benjamin Victor Goldsmith (b. 2008). In 2012, the couple announced they were divorcing after it was alleged Kate had an affair with American rapper Jay Electronica.
Alice Miranda Rothschild (born 1983) is married to Zac Goldsmith, British Conservative Party politician, and the brother of her sister Kate's ex-husband.
James Amschel Victor Rothschild (born 1985), who married Nicky Hilton, socialite great-granddaughter of hotelier Conrad Hilton, in 2015.
Rothschild and his wife were also godparents to a daughter of politician William Waldegrave, who is a godfather to James Rothschild.
Amschel Rothschild owned a summer home at New Alton, New Hampshire.
Amschel Rothschild died in July 1996. He was found hanged at the Hôtel Le Bristol Paris. The death came hours after a meeting, during which plans were discussed for a merger of asset management companies.