Burgess was born in Chelsea, in London, and was the son of Henry William Burgess, landscape painter to William IV, and part of a family of several generations of distinguished artists (see "Family" below). He was educated at "Brompton Grammar School" and, after the death of his father when he was 10 years old, received art training from William Charles Ross, the miniature painter - who had been friend of his father. In 1848 he went to James Mathews Leighs art school in Soho. In 1850 he exhibited a picture at the Royal Academy and in 1849 entered the Academy schools, winning the first-class medal for life drawing. Each year from 1852 until his death, Burgess was an annual contributor to the Academys exhibitions.
Burgess started his career by painting portraits and genre works, before travelling to Spain in 1858, accompanied by his friend and fellow artist Edwin Long - who would become his travelling companion on future painting trips to the country. For the next some thirty years, Burgess was an annual visitor to Spain, often spending days with Spanish peasants, living their life and sharing their food. He also went to Morocco at least once. In 1860, he married Sophia, daughter of Robert Turner of Grantham, Lincolnshire.