He was born in Gateside, Newliston near Linlithgow on 6 October 1742 the youngest son of William Dalzell, a carpenter, and his wife Alice Linn. His father died in 1751 and the young Dalzell then fell into the financial care of his namesake and uncle, the Rev Andrew Dalzell of Stoneykirk but remaining in Newliston under the supervision of Rev John Drysdale of Kirkliston.
His early education was at Kirkliston Parish School, and then he attended Edinburgh University studying to be a minister in the footsteps of his uncle and adoptive father, but he was never licensed to preach.
In 1783 he was a co-founder of the Royal Society of Edinburgh. He is said to have taken his pupils to Glasgow to hear the lectures of John Millar on Civil Law. This brought him into contact with the critical figures of the Scottish Enlightenment, most of which were in Edinburgh.
Curiously his address is given simply as College at this period, as he lived in tied accommodation within Old College.
He died (after a long illness) on 8 December 1806. He is buried in Greyfriars Kirkyard in central Edinburgh.
His burial vault lies against the south wall and is highly defensive in design, being constructed at the height of the graverobbing fears in Edinburgh. His role as Professor of Greek was taken over by George Dunbar who had acted as his assistant for many years.
Dalzell presumably met her through this connection.
