The Verdict: Clear Path
You are very probably a British citizen already: almost every British overseas territories citizen became one automatically on 21 May 2002. Apply for a British citizen passport with evidence of your status before that date.
Section 3 of the British Overseas Territories Act 2002 made everyone who held the status immediately before that date a British citizen, with no application or fee. The only exclusion is a connection solely with the Sovereign Base Areas in Cyprus.
Children born to such citizens since then generally hold both statuses. If you acquired yours after 2002, check your own dates with the Passport Office.
The evidence is a territory passport from that period, a birth certificate showing the connection, or a registration or naturalisation certificate. If you were a BOTC by descent, you are a British citizen by descent and cannot pass it automatically to a child born abroad.