William is quite a common name amongst the Dunkley family but with the specific information you have you can easily obtain his birth certificate and on that it will have the name of his father and mother, the maiden name of his mother and the occupation of his mother.
From there it is easy to trace his father or mother back through censuses - 1891, 1881, and so on back to 1851.
You can obtain a copy of the full birth certificate by sending his full name, date of birth and if you know it place where he was born to: The Registrar General, The General Register Office, PO Box 2, Southport, Merseyside PR8 2JD England. It will cost you £11 and the cheque or money order should be made out to ONS. If you go to the ONS or Public Records Office website there is more information.