Cotman was the son of a hairdresser, Edmund Cotman (1759 - 1843/4), and his wife, Ann Sell (1763 - 1835). He was educated at Norwich Grammar School and moved to London in 1798 to work as an assistant to the publisher Rudolph Ackermann (1764 - 1834). He soon joined the circle of physician and art collector Dr Thomas Monro (1759 - 1833) and attended evening classes at his house in Adelphi Terrace, London, copying drawings from his collection. Cotman first exhibited at the Royal Academy of Arts in London in 1800. He was awarded the large silver palette by the Society of Arts in the same year.
Cotman embarked on many sketching tours in England and Wales between 1800 and 1805. He was an elected honorary member of the Society of United Friars of Norwich and was a leading member of the Sketching Society in London. He continued to exhibit his work at the Royal Academy until 1806 when he returned to Norwich after failing to become a member of the Society of Painters in Water Colours (Old Watercolour Society).
In Norwich, Cotman set up a school for drawing, advertised his services as a portrait painter and held a large exhibition of his work to attract commisions and patrons but his efforts were largely unsuccessful. He was elected vice-president of the Norwich Society of Artists in 1810 and President in 1811. During this time he also exhibited with the Associated Artists in Water-Colours Society and the British Institution in London.
Cotman moved to Yarmouth in 1811 and continued to teach. He also made etchings for various publications including 'Architectural Antiquities of Norfolk' and 'Architectural Antiquities of Normandy' following trips to Normandy in 1817, 1818 and 1820. On returning to Norwich in 1823 he opened another drawing school at his residence and again exhibited at the British Institution in 1823 and 1827. He finally became an Associate of the Society of Painters in Water Colours in 1825 and was elected President of the Norwich Society of Artists for the second time.
Cotman moved back to London in 1834 to take up the post of drawing-master at King’s College School. He remained in the capital until his death in 1842. Henry George Bohn (1796–1884) published Cotman’s 'Liber Studiorum' and his collected etchings 'Specimens of Architectural Remains in Various Counties in England' in 1838.