It is impossible to give a history of the O'Brien clan in a few sentences. With Brian Boru the sept rose to the High Kingship of Ireland in the 10th century. They remain prominent in the history of the country to this day.
Like all the early septs in Ireland, it is possible that our Brien ancestors are not related to the founders of the sept but rather from some of their followers. All we know is that Michael Murphy's mother is named as Maria Brien
Two possible birth records have been found in Ireland for Michael one would put the family in County Wexford, the other in County Cork. However, with an 1833 birth year, it is also possible that the record is not available. Without further information we don't really have a focus for looking for Brien records.
O'Brien Clan History
O'Brien Clan Foundation
O'Brien Family History - Clare Library
O'Brien Dynasty - Wikipedia