Crampton was born in Dublin, Ireland, the son of a dentist. He was a childhood friend of Theobald Wolfe Tone, the United Irishman, and a cousin, on his mother's side, of Thomas Verner, Grand Master of the Orange Order. He joined the army when young and became an assistant surgeon. When he was appointed surgeon to the Meath Hospital in 1798 he was not yet fully qualified, and went on to graduate in Glasgow in 1800. A few years later he also became assistant surgeon at the Lock Hospital, Dublin and also built up a large private practice at his house in Dawson St. He joined Peter Harkan in teaching anatomy in private lectures, forming the first private school of anatomy and surgery in the city.
He became a Fellow of the Royal Society (F.R.S.) in Ireland for a treatise on the construction of eyes of birds, written in 1813. This was later published, with other writings, in the Dublin Journal of Medical Science.
In 1821, together with Sir Henry Marsh and Dr. Charles Johnston, he founded a children's hospital in Pitt St. (now Balfe St.): the Pitt St. Institution. This hospital was the first teaching children's hospital in Ireland or Great Britain. The main objective of the hospital was to treat sick children in one of the poorest parts of Dublin, The Liberties.
He resigned the chief-surgeoncy of the Lock Hospital when he was appointed surgeon-general to the forces in Ireland. He remained as consulting surgeon to Dr Steevens' Hospital and the Dublin Lying-in Hospital. He was three times president of the Dublin College of Surgeons. He was knighted in 1839.
He was always interested in zoological science and played an active part in founding the Royal Zoological Society of Ireland and was many times its president. He was also a member of the Royal Irish Academy.