Do I have dual British/Irish citizenship?

You father born in Northern Ireland is British, being born to a ROI father means he is also Irish at birth (although he may not have applied for his Irish passport )

If then you can prove your GRAND PARENTS ( or father) were Irish citizens at the time of your birth, so a copy of your full birth certificate showing your father, a copy of his full birth certificate showing his Irish parents... Irish citizenship is granted back to Irish grandparents, so not just parent............. so yes you are ( or could claim) both British and Irish citizenship

The Irish Nationality and Citizenship Act considers the entire island as Irish and does not distinguish between born in the ROI or in Northern Ireland.

If either of your parents was an Irish citizen who was born in Ireland, then you are automatically an Irish citizen, irrespective of your place of birth.

However before you can claim Irish citizenship, you must have your birth registered in the Foreign Births Register, which is maintained by the Irish Department of Foreign Affairs and Trade. If you are entitled to register, your Irish citizenship is effective from the date of registration - not from the date when you were born.

http://www.citizensinformation.ie/en/moving_countr...

You are entitled to Irish citizenship as well as British. Get yourself onto the Foreign Births Registry and then apply for an Irish passport. The process is not complex and this is a valuable citizenship as it passes indefinitely through foreign born as long as each successive generation gets onto the register before their own children are born.