Kilburne was educated at Hawkhurst, Kent - his fathers old school. On leaving at the age of 15, he went to London to serve a five-year apprenticeship as a wood engraver with the Dalziel brothers. He was highly regarded by his employers who described him as "industrious and constant" and "one of the most satisfactory pupils we ever had". His time as an engraver served him well, allowing him to develop the accuracy and detail which would enhance his subsequent painting. He remained with the firm for a further year before leaving to take up watercolour and oil painting professionally, quickly becoming one of the most sought after and well-known artists in England.
In June 1862, Kilburne married Janet Dalziel at Old Church, St. Pancras, London. She was the daughter of the painter Robert Dalziel, the brother of his former employers. They had three sons and two daughters - of these his eldest son, George Goodwin Kilburne Jnr., became a very well known painter of animals and figures, principally of sporting subjects. In 1881 the family were recorded as living in Hampstead, London. Janet died in 1882. In 1889 Kilburne married Edith Golightly (34 years his junior), with whom he had a further two children - Edith May (Born 1900) and Constance Ivy (born 1902).