[1] Alexander was born in 1756, in what was at the time known as Anson County in the Province of North Carolina (his birthplace is located near the modern city of Concord).
He earned a bachelor's degree from the College of New Jersey (now Princeton University) in 1776 and was commissioned as a surgeon in the North Carolina Line in 1779.
He served through the American Revolutionary War until 1782, and then practiced medicine for a time near Santee, South Carolina.
On November 25, 1805, Alexander was elected governor by the North Carolina General Assembly and served two one-year terms in that office, declining to run for a third.
[1] He married a daughter of a Colonel Thomas Polk, but the couple was apparently childless.