Alexander was an 1823 graduate of the United States Military Academy along with Lorenzo Thomas and George S. Greene. Alexander was brevetted Major at the Battle of Cerro Gordo on April 18, 1847 and Lieutenant Colonel at the Battles of Contreras and Churubusco on August 20 during the Mexican-American War. Alexander, still 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 March 1855, promoted to Colonel, Alexander was appointed command of the new 10th Infantry.