William Bemis Alexander was born around 1828 in Plymouth, Massachusetts, to Samuel Alexander and Deborah Paty. His father was a carpenter who owned $800 of real estate in 1850. Alexander grew up and attended school in Plymouth before beginning work as a carpenter. He married Mary Frances Atwood in Plymouth on September 18, 1853, and their daughter Ida was born around 1855. The family moved to Boston sometime in the late 1850s.
Alexander enlisted in the Union army on April 22, 1861, and mustered in as a 2nd lieutenant in Company B of the 3rd Massachusetts Infantry the following day. He mustered out on July 22, 1861, when his three-month term of enlistment expired. Later that year, on December 4, 1861, he received a commission as captain in Company E of the 23rd Massachusetts Infantry. He was wounded in the hand and arm at New Bern, North Carolina, on March 14, 1862, and he resigned on December 28, 1862. He applied for a federal pension in January 1863 and eventually secured one.
Alexander returned to Boston after leaving the army. By 1870, he owned $500 of personal property, and he employed at least one white domestic servant. The family moved to Newton, Massachusetts, sometime before 1882. They returned to Plymouth by 1900, and he died of cancer there on February 22, 1900.