Alexander Cochrane was the sixth of the surviving sons of Thomas Cochrane, 8th Earl of Dundonald. The eldest son Archibald Cochrane became the earl and lost the family lands on a series of inventions and investments. Many of the younger sons served in the military or had careers supplying it. The next brother, Charles, served in the army and was killed at the Siege of Yorktown; he had married to Catherine, the daughter of Major John Pitcairn. The third surviving son, John Cochrane, was a paymaster and provisioner to the army and navy. His children included Nathaniel Day Cochrane, who became a rear admiral, and probably of the chess player John Cochrane. The next son, Basil Cochrane, made a fortune supplying the Royal Navy in India. Alexander was the sixth son. The seventh, George Augustus Frederick Cochrane, had an army career and served in Parliament. The youngest son, Andrew Cochrane-Johnstone, was an army officer, colonial governor, politician, and fraudster.