He was a veteran of the American Revolution and a member of the Democratic-Republican Party who served in the Delaware General Assembly.
Alexander was born in 1755 in Virginia into an Ulster-Scots family and moved to New Castle County, Delaware, when he was a small boy.
During the American Revolution, Alexander first served in the state militia but soon after joined the 10th Virginia Regiment of the Continental Army as a surgeon.
After the war, Alexander returned to New Castle County and practiced medicine.
He was able to carry heavily Presbyterian New Castle County, but lost the remainder of the state and the election to Federalist Gunning Bedford Sr.