Need help/advice in tracing Irish ancestry.

My great great grandmother was born in December 1869 in the Walsall District of Staffordshire England. Both of her parents are listed in the 1871 UK Census as having been born in Ireland. I sent for her birth cetificate and it lists her father as Matthew Mooney and Mother as Mary Butter. The father died in the UK in 1879 and the family migrated to the US. I suspect that both Matthew and Mary came to the UK from Ireland sometime in the 1840s or 50s but can't be sure. They probably came through Liverpool. I have found both names in the 1851 UK Census but they were both living in different areas of England. Mary near Liverpool and Matthew in Lancashire. Both of these are possibilities. I did a search for a marriage certificate on the BMD but did not find anything that matched up. I suspect they were married in the UK and possibly not in Staffordshire. Their first child was born in 1864. On the 1900 US Census Mary (the mother) is living with her daughter Margaret (my great grandmother) in Pennsylvania. She was 72 in 1900. I am searching for a death date in the US and possible obit.

Does anyone have suggestions on taking the next step back to Ireland?

I have sent for the other 2 childrens birth certificates and Matthew's death certificate in 1879 but I don't think that any of these will offer any insight into where they came from in Ireland.

I found a Patrick Mooney living in Walsall District who was about 6 years younger than Matthew also born in Ireland, that could possibly be a brother but again, I have no way to be sure.

Any help would be appreciated.

Thanks,

R