Kelley was born in Norwich, Connecticut. He began racing in marathons during his college years. From 1950–54, he attended Boston University, located about a mile from the Boston Marathon finish line. While there, he excelled in team races and would run his first two Boston Marathons, in 1953 and 1954. He finished fifth in the 1953 race before following up with a 7th-place finish the next year. After graduating, he finished 2nd in the 1956 Boston Marathon and made his way onto the U.S. Olympic Marathon team which competed in Melbourne, Australia during the same year. He would go on to win the Boston Marathon outright in 1957 while setting a new course record on the remeasured course.
After his win at Boston, Kelley would win several other marathons, including eight consecutive wins of the Yonkers Marathon in Yonkers, New York. As a result of his record setting performance at Yonkers in 1960, he again found his way onto the U.S. Olympic Marathon team and competed in the 1960 Olympics in Rome. He placed 21st and 19th in the Melbourne and Rome Olympic marathons, respectively.
Kelley is the only male runner to ever win both the Boston Marathon and the Mount Washington Road Race, which he won in 1961. Another runner, Jacqueline Gareau, has also won both. He made the ascent in one hour and 8 minutes 54 seconds, nearly seven minutes faster than the winning times in the three previous years the race had been held, 1936-1938. In September 2014, a larger-than-life sized bronze statue of Kelley and his dog (Brutus) was dedicated in downtown Mystic, Connecticut.
After his career as a runner ended, he went on to a successful career as high school running coach. At Fitch High School in Groton, Connecticut, Kelley coached Amby Burfoot, winner of the 1968 Boston Marathon, and Julia Chase, the first woman to successfully challenge the gender barrier of the Amateur Athletic Union. In addition to coaching, he found work over the years as a newspaper columnist, free lance writer, cab driver and running wear store co-owner.