Cochrane was born in Bridgewater, Massachusetts. His father, John Cochrane, had immigrated from Omagh, County Tyrone in what is now Northern Ireland and his mother, Sadie Campbell, had come from Prince Edward Island, Canada, whence her family had immigrated from Scotland. He was also known as "Black Mike" because of his fiery, competitive nature. Cochrane was educated at Boston University, where he played five sports, excelling at football and basketball. Although Cochrane considered himself a better football player than a baseball player, professional football was not as established as Major League Baseball at the time, so he signed with the Portland Beavers of the Pacific Coast League in .