There’s no central index of church records. You need to know the person’s parish in Ireland. For Co Antrim most RC parish records have bene copied and are held in PRONI, Belfast. Be aware that,outside Belfast and other big cities, many RC churches didn’t keep baptism or marriage records before the 1820s. And RC churches generally never kept burial records at any time.
Statutory records of births, deaths & marriages started in Ireland on 1.1.1864 (1st April 1845 for Protestant and Non-Conformist marriages). Prior to that you are heavily dependant on church records, where they exist. Most early Irish censuses have been destroyed so no help there. You really need to try and find out a little more about the family from sources outside Ireland. (Otherwise it’s a needle in a haystack).
To use church records effectively, because there’s no centralised index, ideally you need to know the ancestor’s townland and parish in Ireland. Good sources are poor law applicationswhich usually contain details of the person’s parish in Ireland, plus other family they may have (Mitchell Library in Glasgow for the Glasgow area; local libraries for many other areas). Check every possible census. Census returns sometimes include exact place of birth, instead of just “Ireland”. Look out also for other Irish people nearby in the censuses. Immigrants often settled near people they knew from home. Check any military records, old diaries, gravestones, family bibles, wills and obituaries. Birth certs for some locations (e.g. Scotland) include place of marriage of parents (save for 1856 to 1859).