The 33-year-old former footballer takes over six months after Gauthier Ganaye left for OGC Nice, with co-chairman and fellow American Paul Conway standing in since January.
Murphy played for DC United and New York Cosmos as well as German club VFL Osnabruck. After retiring from playing, he was a scout for the Cosmos then technical director at DC United.
He told the Chronicle: "I am delighted to be here. I can feel the enthusiasm and energy around the club and the community from the few days I have been here and I am looking forward to my time at the club."