He graduated from Lake Charles High School in Lake Charles, Louisiana in 1919. In 1924 he graduated from the United States Military Academy, receiving his commission in the Infantry.
He participated in all of Sixth United States Army"s campaigns, including the occupation of Japan. He was awarded the for his activities in the Leyte and Luzon Campaigns, and the for reconnaissance under fire at Biak, Leyte, and Manila. He was promoted to Brigadier General at the Leyte beachhead.
He was then selected to be the first Deputy Commandant of the Armed Forces Staff College.
He was chairman of the faculty board and Director of Instruction of the school for the first four classes. Eddleman arrived in Trieste in June 1949 to become Deputy Commander of Trieste United States Troops (TRUST), and Director General, Civil Affairs, Allied Military Government.
He remained in this assignment for over a year. In November 1950 he was recalled to Washington to become Chief of Plans Division, G-3, and later G-3 of the Army.
He remained in that position until 1954.
In May of that year, he assumed command of the 4th Infantry Division, at the time headquartered in Frankfurt, Germany. He then assumed command of the Seventh United States Army in Germany on July 1, 1958. He filled this position until his retirement on March 31, 1962.
After retiring from the Army, he was a Vice President of Universal Match Corporation for four years, later becoming Director and Corporate Representative.
He also was on the Board of Directors of the Army and Air Force Mutual Aid Association from 1962 to 1980, when he was elected Chairman of the Board, a position he filled until 1982. He was buried at Arlington National Cemetery.
Because of his familiarity with joint operations, he was selected by General Douglas MacArthur to become a member of the Joint Operations Review Board from January to June 1946.