Alexander married his first wife, Helen Stevenson, in 1855. On January 27, 1856, not long after their marriage, Helen died during childbirth with their first child. Their child died as well. Alexander attended New York Law School in Poughkeepsie, New York, and was admitted to the Indiana bar in 1855. He served in the Indiana House of Representatives in 1856. He was a member of the Indiana General Assembly. He lost his bid for United States Senator three separate times. In 1860, Conner took part in the Republican Convention that nominated Abraham Lincoln, and was actively involved in his campaign. In 1861, Alexander was appointed the postmaster for Indianapolis, Indiana. Alexander enlisted in the United States Army, Company B, Indiana 132nd Infantry Regiment, on May 18, 1864. He was promoted to Full 1st Lieutenant on May 19, 1864. Promoted to Full Captain on August 1, 1864. He was Commissioned an Officer in the Indiana 132nd Infantry Regiment, and was honorably discharged, on August 1, 1864. Conner was a newspaper publisher, and was part owner of the Indianapolis Journal for several years, but sold his interest in 1870. In 1871, President Ulysses S. Grant appointed Alexander Governor of Idaho Territory, but he declined the office. He was chairman of the Republican Central Committee 1868 to 1872. Alexander decided it was time for a change, and moved his family to Nebraska, in 1872, where he practiced law. Conner served in the Nebraska Constitutional Convention of 1875 and was elected to the Nebraska State Senate serving in 1883 to 1885 and 1889 to 1891, and served as its president for two years. Alexander passed away in May of 1891.

Alexander's second wife was Katherine A. (Willis) Conner. They were married in 1865, and had 3 children together; William Willis Graves Conner (1866 - 1930), Sarah Katherine Conner (1868 - 1963), and Elizabeth Conner (1871 - 1873). Elizabeth died at a very young age, and Sarah never married, or had any children.