Burke was born in Ballyduggan, near Loughrea, County Galway, the third son of William Malachy Burke (1784–1853) a barrister, and Anna Maria, only daughter of John Blake of Neirfield. Both the Blake"s and the Burke"s were old Galway families, the Burke"s been descended from Richard Mór de Burgh, 1st Baron of Connaught (died 1242) and the Blake"s one of the Tribes of Galway.
After receiving his medical education at Saint George"s Hospital, London, in 1842, he passed at the London College of Surgeons (Membership of the Royal College of Surgeons). He became a Licenciate (LKQCP) in Ireland in June 1847, and on 19 October 1863, a Fellow of the King"s and Queen"s College of Physicians in Ireland. He served as a physician at Steevens Hospital, Dublin, and he was also Physician in Ordinary to the Lord Lieutenant of Ireland, for the years 1866-1868 and 1874-1876.
He also served as a Consultant Physician at the National Eye and Ear Infirmery, Saint Stephen"s Green.
After serving as Medical Superintendent in the General Register Office from 1864 to 1876, he succeeded Mr. Donnelly as Registrar General for Ireland.
He served from 1876 to 1879, during which period he effected important improvements in the system of registration in Ireland. He died childless on 13 August 1879 from pleuro-pneumonia, at his residence at 88 Saint Stephen"s Green.
He was interred in Mount Jerome Cemetery.
Burke was considered to be "a very amiable man and was much liked in the large social circle in which he moved.".
He was a member of the Pathology Society and the Geological Society, Dublin. Of the British Medical Association. A Fellow of the British Meteoroligal Society.
And was a Visiting Physician to Steevens Hospital, Dublin.
