Son of C.H. Kellaway, F.R.S., a distinguished scientist, Frank Kellaway was born in London and educated in Australia at Geelong Grammar School and Melbourne University. He served as a lieutenant, R.A.N.V.R., in World War Two. His various jobs included librarianship and teaching at the Preston Institute of Technology.
Apart from fiction and poetry, Kellaway wrote verse libretti for the music of George Dreyfuss (q.v.) in operas performed in Australia and abroad. He participated in readings of his work as early as 1947 with Alan Marshall and others; he continued to read at various Victorian venues. His papers include correspondence with Gwen Harwood (q.v.), Hilary Cohen (q.v.), Tony Riddell (q.v.) and Barry Dowling (q.v.). Bruce Dawe (q.v.) has said that he had been influenced in his early writing by Kellaway, a writer in the vernacular.