Goodenough was the youngest son of Samuel Goodenough, bishop of Carlisle, by his wife, Elizabeth, eldest daughter of Dr. James Ford, physician extraordinary to Queen Charlotte, was born at Baling, Middlesex, on 3 April 1785.
In 1801 he obtained his election to Christ Church, Oxford, where he took honours in Easter term 1804, and graduated B.A.
Having taken orders, Goodenough became tutor and censor of Christ Church, and in 1810 was appointed curate of Cowley, Oxford.
In Michaelmas term 1817 he was appointed select preacher to the university, and in the following year was instituted vicar of Warkworth, Northumberland.
In 1819 Goodenough was appointed headmaster of Westminster School and subalmoner to the king, in succession to William Page.