Hamilton "Ham" Williams Greene was born May 19, 1904 in Boston, Massachusetts. His father, Russell Thomas Greene, Jr., was born 1874 in New Jersey. His mother, Winifred Williams, was born 1873 in New York of English ancestry. His parents married in 1899. They had six children, of which he was the third born. His older sisters Mary and Eleanor were born in 1901 and 1903, and his younger sisters Sybil, Winifred, and Ruth were born in 1909, 1911, and 1914. His father was a school administrator and mathematics teacher, who graduated Harvard in 1897 and authored a textbook on accounting.