Billy Richardson was the son of Matthew Richardson, a former chairman of the St. James’ Gate club. He played junior for Mountpleasant and made his first appearance for Bohemians in the League of Ireland in 1942–43. Richardson was eventually selected to play on the 1948 Irish Olympic football team alongside Bohemians teammates Paddy Kavanagh, Donald Lawler, Emmett McLaughlin, Bobby Smith and Brendan O’Kelly; the team would go on to lose in its only match to the Netherlands by a score of 3-1. He later remained with Bohemians until the 1949–50 season, appearing in a total of 159 matches with them in 7 seasons.