The Verdict: Clear Path
Born on the island of Ireland before 2005, you are an Irish citizen, or entitled to be one if born in Northern Ireland. Apply for an Irish passport with your civil birth certificate.
Everyone born on the island before 1 January 2005 holds this entitlement, whatever their parents' nationality. The one exception is a child whose parent was a foreign diplomat entitled to diplomatic immunity at the time of the birth. If you were born in Northern Ireland, applying for the passport is how you take it up, and it does not affect British citizenship: you can hold both.
This is a passport application, not a citizenship application. For a first passport you need your civil birth certificate, proof of identity and witnessed photographs.
Your own children born outside the island are Irish citizens automatically, because you were born on it. Your grandchildren born abroad can claim through the Foreign Births Register.