Murphy was born in San Francisco, California, in September 1894. Both his parents were Irish immigrants: his father from County Wexford and his mother from County Mayo. He was orphaned early — his mother died in 1897 and his father in July 1906, shortly after the San Francisco earthquake — and was raised first by his paternal uncle in San Francisco, then by his maternal aunt and her husband, Judge Martin O'Donnell, in Vernon, California. In Southern California, Murphy attended Huntington Park High School and became an accomplished motorcycle rider and mechanic. Before completing his education he opened a garage with a friend, serving motorcycle and automobile owners from the Los Angeles area.
Murphy entered racing not as a driver but as a riding mechanic, a role common in the era when racing cars carried both a driver and a mechanician. He rode with Eddie O'Donnell's Duesenberg at the 1916 Corona road race, where the car averaged 85 miles per hour. Over the following years he gained experience riding alongside some of the leading American drivers of the period, including Ralph DePalma, Harry Hartz, Eddie Rickenbacker, Peter DePaolo, and Tommy Milton.
After the First World War, Murphy's transition to driver came through the influence of Tommy Milton, Duesenberg's lead driver, who arranged for Murphy to receive a factory car for the inaugural race at the Beverly Hills Speedway on February 14, 1920. Murphy won the race, to the surprise of most observers, and went on to become a regular winner on the American circuit.
In 1921, as part of a Duesenberg team backed by French immigrant Albert Champion, Murphy entered the French Grand Prix at Le Mans. Competing against strong opposition from Sunbeam — including the young Henry Segrave — and Ballot, Murphy won the race in an American car, becoming the only American to have achieved that feat. The next comparable achievement would not come until Dan Gurney won the 1967 Belgian Grand Prix forty-six years later. Murphy's riding mechanic in the French Grand Prix was Ernie Olsen, who also accompanied him in the 1922 Indianapolis 500.
In 1922, Murphy returned to America and won the Indianapolis 500 in the Le Mans-winning Duesenberg, modified for the occasion and fitted with a Miller engine. That year he also won the National Championship. Murphy continued to race in Europe: in 1923 he placed third in the Italian Grand Prix at Monza, competing for Los Angeles race car builder Harry Miller. He ranked second in the 1923 National Championship.
Murphy's 1924 season was successful on the American circuit. He finished third at Indianapolis and by late September had accumulated enough points to hold an unassailable lead in the National Championship standings.
On September 15, 1924, Murphy agreed to compete in a dirt-track race at the Syracuse, New York fairgrounds — a surface on which he was not regarded as a specialist — as a favour to a promoter friend. Late in the race, on lap 138 of 150, his car slid and struck the inside wooden rail. A section of the rail penetrated Murphy's chest, killing him instantly. He was three days past his thirtieth birthday.
Murphy was buried at Calvary Cemetery in East Los Angeles, in the O'Donnell family plot. In an unprecedented decision, the American Automobile Association's Competition Board awarded the 1924 National Championship to Murphy posthumously.
Murphy's victory at the 1921 French Grand Prix stands as one of the most remarkable achievements in early American motorsport: an American driver, in an American car, defeating the leading European manufacturers on their home ground. His combined record of a Grand Prix win, an Indianapolis 500 victory, and two national championships in a career spanning less than five years placed him among the foremost racing drivers of his era. He was inducted into the Auto Racing Hall of Fame in 1964 and the Motorsports Hall of Fame of America in 1998.