Popham was born at Littlecote House in Wiltshire, the son of Sir Francis Popham and Anne Gardiner Dudley, and grandson of Sir John Popham and wife Amy Games. He was educated at Balliol College, Oxford, and admitted to the Middle Temple in 1622. He was a prominent figure and Justice of the Peace in Somerset. In April 1640 he was elected Member of Parliament for Bath in the Short Parliament. He was re-elected MP for Bath for Long Parliament in November 1640. He also fought in the Parliamentary army with the rank of colonel and had a garrison stationed at Littlecote House. In 1654 he was elected MP for Bath again in the First Protectorate Parliament. He was elected MP for Wiltshire in the Second Protectorate Parliament and for Minehead in the Third Protectorate Parliament. In April 1660 he was elected MP for Bath again in the Convention Parliament. After the restoration of the monarchy, he made his peace with Charles II and entertained him to a "costlie dinner" at Littlecote. He was re-elected MP for Bath in 1661 to the Cavalier Parliament.
Popham married Letitia Carre, daughter of William Carre of Ferniehurst, Scotland, half brother to Robert Carre, favourite of King James I. His daughter Letitia (d. 16 March 1714), one of eight children, married Sir Edward Seymour, 4th Baronet, who served as Speaker of the British House of Commons. His son, Sir Francis Popham (d. 28 August 1674), of Littlecote, Wiltshire, married Helena Rogers and had a daughter Letitia Popham (d. 1738), married to Sir Edward Seymour, 5th Baronet. Alexander's eldest daughter Essex Popham married on 17 August 1663 John Poulett, 3rd Baron Poulett and had issue.
This Alexander Popham is not to be confused with his nephew Alexander Popham, son of Alexander's brother Edward Popham, who was born deaf and was taught to speak by two scientists, John Wallis and William Holder. He is considered to be one of the earliest cases of a born deaf person learning to talk. 
- Popham, Frederick William, “A West Country Family: The Pophams since 1150” (privately printed, 1976)
- Cobbett's Parliamentary history of England, from the Norman Conquest in 1066 to the year 1803 (London: Thomas Hansard, 1808)