Stephanie,
Kain (and it’s many variants eg Kane, Cain, O’Kane etc) is a very common name in the Coleraine area.
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
https://www.nli.ie/en/family-history-introduction.aspx
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If Bernard was RC, Killowen is part of the RC parish of Coleraine. Unfortunately their baptism records only start in 1843 and so if Bernard was born there c 1811, there won’t be a record of his birth. The Church of Ireland’s records start in 1824. If Presbyterian then there are a number of different churches the family might have attended.
The 1831 census of Killowen has survived. There are 11 Kane and 3 O’Kane households in the parish. They were a mix of RC, Church of Ireland and 2 Presbyterian households. If you know Bernard’s denomination that should help narrow the search for the right family. Note that only heads of household were listed. There are no childrens names.
http://www.census.nationalarchives.ie/search/