Mitchell is born in Dublin on April 13, 1780. His family moves to Belfast while he is a child. He receives his formal education at Belfast Academy where he excels in mathematics. He begins to notice that his eyesight is failing. By the age of 16 he can no longer read and by the age of 22 he is completely blind.
Undeterred, Mitchell borrows £100 and starts up a successful business making bricks in the Ballymacarrett area of Belfast. This enables him to start building his own houses and he completes approximately twenty in the city. It is during this period that his talent for inventing comes to the fore and he fabricates several machines for use in brick-making and the building trade.
Mitchell patents the screw-pile in 1833, for which he later gains some fame. The screw-pile is used for the erection of lighthouses and other structures on mudbanks and shifting sands, including bridges and piers. His designs and methods are employed all over the world from the Portland, Maine breakwater to bridges in Bombay. Initially it is used for the construction of lighthouses on Maplin Sands in the Thames Estuary in 1838, at Fleetwood Lancashire (UK) Morecambe Bay in 1839 and at Belfast Lough where his lighthouse is finished in July 1844.
In May 1851 Mitchell moves to Cobh to lay the foundation for the Spit Bank Lighthouse. The success of these undertakings leads to the use of his invention on the breakwater at Portland, the viaduct and bridges on the Bombay, Baroda and Central India Railway and a broad system of Indian telegraphs.
Alexander Mitchell dies at Glen Devis near Belfast on June 25, 1868 and is buried in the old Clifton graveyard in Belfast. His wife and daughter predecease him.