Bunney was from London and by his early twenties was painting, exhibiting and selling as an amateur artist, while working in his family's stationery business. Late in 1855, he began attending evening art classes at London's Working Men's College and probably soon after took a new job as a clerk at John Ruskin's publishers Smith, Elder & Co. Bunney benefited from the art classes so much that he had left the publisher's employment by early 1859 and had become a professional artist, working as a drawing tutor and selling paintings.
At the college, Bunney had met John Ruskin, who was then its drawing master, and in 1859 and 1860 Ruskin sponsored Bunney's first two tours in Europe. It was again Ruskin's idea to send Bunney to Italy in 1863. Bunney and his wife lived first in Florence but they and their children moved to Venice in 1870. In Italy, Bunney received commissions from Ruskin for various projects, including the Guild of St George, but he remained independent of Ruskin and the Guild and fulfilled numerous commissions for other patrons. Bunney died in Venice in 1882.