Conte was educated in public schools and graduated from Dickinson High School.
After graduation, he worked at a number of jobs, including Wall Street messenger and pianist in a swing band. In the summer of 1935, Conte was waiting tables at a Connecticut resort when his employer insisted that he take part in the resort's amateur stage production. Reluctantly, Conte agreed to take the acting assignment in order to keep his job. It was a choice that changed his life. Attending one of the performances were several leaders of the renowned Group Theater in New York City, including Elia Kazan and Robert Lewis, who immediately offered Conte a scholarship to attend their acting workshops at the Neighborhood Playhouse in Manhattan. Conte laughed off the offer, saying he had little interest in the theater. His curiosity piqued, however, he attended the Group Theater's now-legendary performance of Clifford Odets's Waiting for Lefty--the first professional play he had ever seen--and was immediately transformed. "I don't know what happened to me, " he later told the New York Times, "but I cried, and all of a sudden I wanted to be an actor more than anything else. I rushed backstage and asked them to give me another chance at the scholarship. " Lewis happily complied, and Conte was on his way. Conte became deeply committed to his newfound goal in life, gaining a reputation for taking the craft of acting seriously, which earned him high marks with theater directors and producers. After several years at the Neighborhood Playhouse in New York, he began to get roles off and on Broadway. His big break came with a significant part in the Broadway hit Jason (1942). Critic George Jean Nathan called Conte the outstanding young actor of the year, and other notices were equally glowing. He followed that success with a widely praised performance in another Broadway production, The Family (1943).
As his career was picking up speed, World War II began, and Conte was drafted into the army. His military duty consisted largely of serving in the entertainment corps, and he was released after less than a year because of medical problems. Twentieth Century-Fox signed him to a contract in 1943 and required only that he change his name to Conty, which was more clearly pronounceable and less ethnic. Conte compromised by changing his first name to Richard. He had his own reasons for changing his name--he hoped to avoid being typecast as a "Mediterranean type. " His classic Italian features betrayed him, however, and Hollywood producers cast him accordingly. Conte did not complain; the roles were usually challenging, and the films were generally high-quality productions. Indeed, Conte rarely complained about anything. He gained a reputation as a hardworking, well-focused actor and was rewarded for his dependability.
Having started his film career during World War II, he found himself in a number of war films during his first years in Hollywood. His first major film, Guadalcanal Diary (1943), was well received and is considered to be several cuts above the usual World War II action movie; Conte's performance as Captain Davis received many plaudits. He next appeared in A Bell for Adano (1945), a film about Americans who capture an Italian town and make peace with the inhabitants. This well-crafted picture was immediately embraced by the American public because its uplifting end-of-the-war message. A Walk in the Sun (1945), another war picture with an ensemble cast, had quite a different message. Lewis Milestone, who had directed the fiercely antiwar All Quiet on the Western Front (1930), delivered a less polemical view in Walk in the Sun, but the film directly challenged the celebratory view of war common in American cinematography. Conte played a wise-cracking Italian-American soldier participating in the liberation of Italy. Unlike Adano, however, the film does not end in reconciliation: all that the Allies have won is a small farmhouse housing an assortment of Nazis, and the casualties on both sides are substantial. Although the critics praised the film's somber message, the public was more interested in healing the wounds of war than in circumspection. After Conte turned in yet another effective and understated performance in Joseph Mankiewicz's crime thriller, Somewhere in the Night (1946), the studio heads at Fox decided to try him in more challenging roles. Their enthusiasm for Conte's work was no doubt heightened by the ever-increasing fan mail he was receiving as well as the generally successful track record of the films in which he appeared. In Cry of the City (1948), a gritty film noir directed by Robert Siodmak, Conte gave what was described as "a blood-chilling performance" in the role of a cynical, heartless criminal who is chased down by a police lieutenant, played by Victor Mature. The role gave Conte a chance to stretch beyond the generally upbeat characters he had played in the past. Following the success of Cry of the City, Conte's status as a leading man was assured. For the next few years he continued to star in very good, although not quite great movies, most of which were successful at the box office. In Call Northside 777 (1948), another crime thriller, Conte played a Polish American wrongly accused of murdering a policeman. In this film he turned in a performance that became his trademark--intensely focused with complex character shadings. Over the next few years Conte continued to appear in tough-minded, well-crafted crime films such as Thieves' Highway (1949), The Sleeping City (1950), The Blue Gardenia (1953), and New York Confidential (1955). In all of them he played characters caught within moral dilemmas, whichever side of the law they happened to walk on. In 1956, Conte asked for a change of pace and was given a part in the lighthearted domestic comedy Full of Life. Although he once again played an Italian, he was glad to have the opportunity to prove his skills as a comic actor. Appearing opposite Judy Holliday, he received wide praise for his effort, and the film was a box office smash.
Full of Life marks both the pinnacle of Conte's career and its decline. Although he continued to take leading roles into the 1960's, none of the parts he was given matched the quality of those he had enjoyed in the past. In part, the fading of the crime drama left him without the roles for which he was noted, and he was not able to go much farther in comedy.
During the 1960's he picked up television parts, none of them very memorable. He also appeared in supporting roles in such films as Who's Been Sleeping in My Bed? (1963), Synanon (1965), and Hotel (1967). Once, in 1969, he tried his hand at directing in the uninspiring Operation Crossbow. Fittingly, Francis Ford Coppola paid tribute to Conte's legacy in crime drama by casting him in the role of the mobster Don Barzini in The Godfather (1972).
He married his sweetheart of several years, Ruth Strohm, a New York actress. They had one child. Just as Conte's career was slowing down, he and his wife divorced in 1963.