Hello Thomas
I put your g grandmother'sname into a search engine but the name of Murphy in Cork brought back too many hits .As you haven't a parents name to refine your search it will prove a little difficult to find her.
However, here's where I would suggest you start.
Go to www.castlegarden.org and try to find out if she came to the us through this portal.
If you don't find her there go to www.ellisisland.org and perform a search there.
Try www.familysearch.org
the church parish records are on two websites here in Ireland. www.rootsireland.ie which has the eastern half ot the Cork records, and www.irishgenealogy.ie which has the western half of the records.
If I could give you one piece of advice as you start your search is to KEEP AN OPEN MIND.
Most people who emigrated during the 19th century were uneducated and didn't have a birth certificate. They hadn't a clue as to what age they were, and when asked in port of arrival took a wild guess. So widen your date range for your searches.
The state records began to be kept here in 1864 and her birth would have been registered by the state. These records are accessed through www.groireland.ie However they can't just do a trawl for Catherine Murphy . They usually require a little more information such as parents names or parish.
Hope this is of some use to you,
Good luck in your search,
regrds,
Anne
