Price was born Dennistoun Franklyn John Rose-Price in Twyford in Berkshire, the son of Brigadier-General Thomas Rose Caradoc Price CMG DSO and his wife Dorothy, née Verey, daughter of Sir Henry Verey, Official Referee of the Supreme Court of Judicature. He attended Copthorne Prep School, Radley College and Worcester College, Oxford. He studied acting at the Embassy Theatre School of Acting.
Price made his first appearance on stage at the Croydon Repertory Theatre in June 1937, followed by a London debut at the Queen's Theatre on 6 September 1937 in Richard II.
He served in the Royal Artillery from March 1940 to June 1942 during World War II but quickly returned to acting after discharge, appearing with Noël Coward in This Happy Breed and Present Laughter and later as Charles Condomine in Blithe Spirit, which he later named in Who's Who in the Theatre as one of his two favourite parts along with the title role in André Obey's Noah.
His first film role was a good one, in A Canterbury Tale (1944).
Price impressed Gainsborough Pictures who put him under contract. According to Brian MacFarlane, Price was "mercilessly used by Gainsborough [Pictures] in one unsuitable role after another" in this period.
He was given a support role in A Place of One's Own (1945) starring James Mason. British National borrowed him for The Echo Murders (1946), a Sexton Blake film, then he was fourth-billed as the villain in a Gainsborough melodrama, Caravan (1946) with Stewart Granger and Jean Kent, playing the type of villainous part that had made James Mason a star (and was no longer interested in playing). It was a huge success.
He was a villain again in Gainsborough's The Magic Bow (1946) with Granger and Kent. Two Cities Films used him in one of their melodramas, Hungry Hill (1947). Gainsborough used him in villainous roles in Dear Murderer (1947), Holiday Camp (1947), Jassy (1947), and Master of Bankdam (1947).
He made two for Bernard Knowles, supporting Margaret Lockwood in The White Unicorn (1948), and a comedy Easy Money (1948). He followed this with a thriller, Snowbound (1948), and a crime melodrama Good-Time Girl (1948). In 1948 British exhibitors voted Price the tenth most popular British actor at the box office.
Price was promoted to starring roles. He was given the title role in The Bad Lord Byron (1949), which was a huge box office flop that helped kill off the Gainsborough melodrama. More popular was Kind Hearts and Coronets (1949), for Ealing Films; Price played the suave serial murderer Louis Mazzini alongside Alec Guinness.
Price was in a wartime drama The Lost People (1949). He was a guest judge on a BBC radio broadcast of the Piddingtons show in 1949. His role was to represent the eyes of listeners as the Piddingtons performed their telepathy act in the Piccadilly studios, and in the Tower of London. He was ensuring that no cheating was going on and overseeing the telepathy tests as a witness.
Price loaned to Associated British Picture Corporation to make two films: the musical The Dancing Years (1950), a sizeable hit; and the thriller Murder Without Crime (1950), less popular.
Back at Rank he was a villain in The Adventurers (1951) and was borrowed by 20th Century Fox in I'll Never Forget You (1951).
Price played the lead in Lady Godiva Rides Again (1951), and after a cameo in The Magic Box (1951) he was in a comedy, Song of Paris (1952).
Price supported in The Tall Headlines (1952) and had the lead in some Bs: Noose for a Lady (1953), Murder at 3am (1953) and Time Is My Enemy (1954). In "A" pictures he was now a supporting actor: The Intruder (1953), For Better, for Worse (1954), That Lady (1955), Oh... Rosalinda!! (1955), Private's Progress (1956), Charley Moon (1956) with Max Bygraves, Port Afrique (1956), A Touch of the Sun (1956), Fortune Is a Woman (1957), The Naked Truth (1957), Danger Within (1959), and I'm All Right Jack (1959). He was top billed in Don't Panic Chaps! (1959).
In the 1950s, Price appeared in London and New York in new plays and revivals of classics. It has been suggested that he was the first name actor on television to play a "more or less overtly gay role" in Crime on Our Hands (1954). In 1957, he made his debut in South Africa in lead roles in Separate Tables.
As a radio actor, Price was the original "No. 1" in charge of the crew of HMS Troutbridge in the first series of the long-running radio comedy series The Navy Lark in 1959, but was unable to continue the role in the second series because of other work commitments; he was replaced by Stephen Murray. His film appearances from this period included Tunes of Glory (1960) and The Amorous Prawn (aka The Playgirl and the War Minister, 1962). In Victim (1961) he portrayed one of several characters being blackmailed because of their (then illegal) homosexuality.
In the BBC television series The World of Wooster (1965–67), Price's performance as Jeeves was described by The Times as "an outstanding success". Working with Ian Carmichael as Bertie Wooster, this now almost completely lost series was based on the novels and short stories of P.G. Wodehouse. He also appeared in an episode of The Avengers.
In 1967, Price was declared bankrupt; he attributed his financial distress to "extravagant living and most inadequate gambling". He then moved to the tax haven island of Sark, which coincided with an escalation in his alcoholism. Towards the end of his life, Price appeared in a series of horror films such as Twins of Evil (1971), Horror Hospital (1973) and Theatre of Blood (1973), including five movies directed by Jesús Franco. One of his last film appearances was a star-studded version of Alice in Wonderland (1972) with Ralph Richardson, Robert Helpmann, Peter Sellers and Dudley Moore, among others. On television he had recurring roles in the ITC series Jason King (1971) and The Adventurer (1972).
Price died of heart failure complicated by a hip fracture in Guernsey at age 58 in 1973. He is buried on Sark.
In the book British Film Character Actors (1982), Terence Pettigrew wrote that Price's most successful screen characterisations were "refined, self-centred, caddish and contemptuous of a world inhabited by inferiors. Everything about him was deceptive. He could be penniless and still manage to look as if he owned the bank. But behind all that grand talk and lordly ways, there skulked, in his characters, the most ordinary of shabby, grasping souls."
Price was married to the actress Joan Schofield from 1939 to 1950. They had two daughters. Price was homosexual and the marriage was an attempt to lead a 'conventional' life, since homosexual activity was a criminal offence in England and Wales until 1967. In April 1954, he tried to commit suicide by gas in a London guest house. Public sympathy led to a revival of his career and the offer of film roles.
However, his private life, which included heavy gambling and an increasing reliance upon alcohol, began to affect his health, looks and career.