Well that gives you a choice of Church of Ireland, Methodist or Presbyterian. You can see what records exist for each of those denominations in Clones on the PRONI website:
http://www.proni.gov.uk/index/research_and_records_held/catalogues_guid…
Unfortunately no censuses exist in Ireland prior to 1901 (save for a few fragments). Statutory recording births & deaths didn?t start till 1864. So you?ll probably need to rely on church records to develop your knowledge of the family. The records are not all on-line and you may need to research them in PRONI in order to get the information you need.
Griffiths Valuation for 1861 records just one Wm Monaghan in Clones. He had a house, yard & small garden in Fermanagh St, Clones. (Plot 64 on Griffiths). You can follow through to see how long he stayed at that location by using the revaluation records, in the Valuation Office in Dublin. (Not on-line unfortunately).
http://www.askaboutireland.ie/griffith-valuation/index.xml
There was a James Monaghan in Fermanagh St in the 1901 census. However he was a shop assistant in a pub/general store and so probably unconnected. He was also RC.
http://www.census.nationalarchives.ie/pages/1901/Monaghan/Clones_Urban/…