Murphy was raised in San Francisco. His mother died in the April 18, 1906 earthquake and he then lived with his uncle in Los Angeles. Murphy learned to ride a motorcycle and because of the nature of motorcycles in his day, he soon became an expert mechanic.
On his 21st birthday, he joined the Duesenberg team as a riding mechanic, by 1921 he was a driver and won the opening championship round on the boards of Beverly Hills. That year was also his debut in the Indianapolis 500 where he finished fourth.
Murphy and the four-car Duesenberg team then headed to France to race in the French GP at Le Mans. He crashed on a practice run. On race day he was in the hospital just two hours before the start of the race. He required assistance to get into his race car, but he then proceeded to decimate the field and crossed the finish line as the Grand Prix winner.
In 1922 he qualified on the pole for the Indianapolis 500 with the same car in which he had won the French Grand Prix. Murphy went on to win the race by over three minutes. He dominated the race by leading 153 of the 200 laps. Murphy added six more victories that year to earn him his first National Championship crown.
In 1923, Murphy placed second in the National Championship, even after missing several races to go to Europe and race for Los Angeles race car builder, Harry Miller. He placed third in the Italian Grand Prix at Monza that year.
Murphy's success continued in 1924. He finished third in the Indianapolis 500, and by the last weeks of the season, he had accumulated an unbeatable lead in points toward the National Championship. At the Syracuse, New York fairgrounds dirt track Murphy died after crashing into the outside fence. Murphy followed Joe Boyer and Dario Resta as the third Indianapolis 500 winning driver to lose his life in two tragic weeks.