Alexander was born in Haymarket, Virginia[1] and an 1823 graduate of the United States Military Academy at West Point, New York – along with Lorenzo Thomas, Alfred Mordecai and George S. Greene.

Alexander, as a brevet lieutenant colonel, led the 8th Infantry, when it became the first garrison stationed at Fort Bliss in the El Paso, Texas, area, from January 1854 through March 1855.

In time of war, in the presence of an enemy, remember that these colors are far more precious than life itself; follow wherever they may lead; gather around them in moments of peril, and rather than see yourselves deprived of them, die like faithful soldiers beneath their cherished fold.

Alexander served during the American Civil War as Acting Assistant Provost Marshal General, Superintendent of Volunteer Recruiting Service, and Chief Mustering and Disbursing Officer for the State of Missouri, with headquarters at St. Louis from May 1, 1863, to Apr.

After the war, he was in command of 10th Infantry at Fort Snelling, Minnesota from May, 1866 until he retired from active service on February 22, 1869,[5] under the Law of July 17, 1862, as he was over "the Age of 62 Years."