Irish civil records start in 1864 for all citizens. They are indexed by year and available from the LDS on microfilm. The index are against registration districts NOT counties. Thus, one must know the registration districts for Co. Clare:
Please note that these films are NOT the IGI although some entries in the IGI are abstracted from these records. Some FHL volunteers get confused when patrons ask about these.
I would look for the marriage (1870 onwards) and work from there. The marriage record will give townland of residence and names of fathers. Once you have townlands then you can go to parish records to work back to previous generations.
See the excellent, free, on-line article on these records at the hot link below.
Kathy Flynn, CGRS