Johnston was born at Detroit, Michigan, in 1859. He received his preparatory education at Amherst, Massachusetts, and at Ann Arbor High School. He enrolled in the literary department at the University of Michigan in approximately 1877 and received a bachelor's degree in 1881. He received his doctor of medicine degree there in 1883. While studying at Michigan, Johnston also played at the halfback position for the 1879 Michigan Wolverines football team, the first football team to represent the University of Michigan. He also played for the 1880 team. He was also the vice president of the Student Athletic Association.