William’s birth was long before the start of statutory birth registration in Ireland (1864) so you won’t find a birth certificate for him. You might find a baptism if the records have survived, and you can locate his exact religion and parish or townland. Looking at Griffiths Valuation, there are about 20 May entries for Co Armagh but only one for Co Monaghan (in 1860), so obviously Armagh seems more likely to be his county of origin.www.askaboutireland.ie/griffith-valuation/index.xml
I would try to find William’s marriage cert in England. From that you should get his father’s name, which I would then search against Griffiths. If you find a possible match then that could form the basis for a search of parish records to lcate his birth and that of any siblings.
Irish censuses pre 1901 have been destroyed so you won’t be able to find his family through that route.