Alexander's family had come to America as refugee French Huguenots in 1685. Arriving first in Maryland, part of the family settled in Prince William County, Virginia. Alexander was born there in 1761, the son of Cuthbert Bullitt and Helen (Scott) Bullitt. His father was a leading lawyer in local and colonial affairs. Alexander's early schooling was directed at making him a lawyer. But his uncle Thomas Bullitt was a pioneer and military leader, involved with western exploration. In 1783 Alexander followed in his uncle's footsteps and moved to Kentucky.
Bullitt settled first in an area that is now Shelby County but soon found the area too isolated, and subject to Indian attacks. His uncle Thomas had surveyed a town site in 1773 at the Falls of the Ohio, and in 1800 the Virginia legislature named it Louisville. Alexander bought 1000 acres about 9 miles south of there and began his farm again. He named it Oxmoor, after the fictional farm in Tristram Shandy. In 1784 he made his first steps in Kentucky politics. He became an officer in the local militia, and met with others at Danville in a convention that first proposed a separation from Virginia.
In August 1785, his neighbor, Colonel William Christian, who had been developing a neighboring property, brought his wife Anne (née Henry) and family from Virginia. Two months later Alexander married their young daughter Priscilla. While this improved his political connections (her uncle was Patrick Henry), they were a devoted couple. They built Oxmoor, starting the main house in 1787, and there they grew tobacco, hemp, and corn. They raised a family of four: William Christian, Anne, Helen Scott, Cuthbert on the farm.
After Colonel Christian was killed in a skirmish with Indians in 1786, Bullitt became a major in the militia. He was made a trustee of the town of Louisville. In 1792 when the idea of Kentucky statehood was accepted, he was a delegate to the Convention (again in Danville) where he was on the committee that drafted the first state Constitution. With the new constitution in place, in June he was one of twelve men elected to the State Senate. He was elected its speaker, and served there until 1800.
When a second state constitution was sought, in 1799, Bullitt presided at the convention that wrote it. This new constitution created the office of Lieutenant Governor and he was elected to that office in 1800 serving under Governor James Garrard.
In 1804 Bullitt returned to private life and the full-time development of Oxmoor.
After his first wife died in 1806, Bullitt remarried, to widow Mary Churchill Prather. They had two children, Thomas James and Mary (Polly). Henry Clay's last court case, concerned the will of Mary (Polly) Bullitt that was challenged by Alexander Scott Bullitt's two sons.
He died at his home in 1816 and is buried in the family graveyard in Louisville. Bullitt County, Kentucky was named in his honor.