The first step is to get the relevant application forms. The Citizens Information website at http://www.citizensinformation.ie/en/travel_and_recreation/travel_abroad/passports/applying_for_or_renewing_an_irish_passport.html describes the basic process for application - on this it notes that because I live outside of the Republic of Ireland, I must obtain an APS2 form, but this is unfortunately not available online.
The advice on this Citizens Information page states that "If you are living abroad you can apply in person or using ordinary or registered post to your nearest Irish embassy or consulate." Living in Scotland, the nearest consulate to me is in Edinburgh, and so a quick search on Google reveals the relevant passport page on its website at https://www.dfa.ie/irish-consulate/edinburgh/passports/how-to-apply-for-a-passport/. On this I was advised to contact them by email, with my home address, to request copies of the required number of APS2 forms.
Having done so, within an hour I received an email from Edinburgh notifying me that the required form will be sent by post to me this evening. We're under way! I am also advised that at present, the turn around time for successful applications is 7 weeks, although a priority service is available at additional cost. I'm happy with the seven weeks, thanks. :)
Under the Documents tab on this same page, there is a list of accompanying document types that I will need. As I am applying for an Irish passport for the first time, I am advised that I will need to obtain a copy of my long form birth certificate from the General Register Office of Northern Ireland to prove that I was indeed born in Northern Ireland. This can be done via the GENI website at https://geni.nidirect.gov.uk/. (NB: If I was a woman I would also need to obtain a civil marriage or civil partnership certificate, if my name was now different to that shown on my birth cert).
On the GENI website, rather than use the top section I would normally use to Search registrations for genealogical research, I instead need to order a birth cert from the Order Certificates section. This can all be done online, which takes a couple of minutes, as I step through various pages, providing my name, date and place of birth, and the details of my parents. Once completed, I am charged £15 all in. As I am not paying for priority postage, I am notified that it will be sent out to me by second class post.
Note that I am applying to the GRONI in Belfast because of my Northern Irish birth, but if I was from the Republic of Ireland, I would need to apply to the separate General Register Office of Ireland to obtain the southern equivalent - to do so, visit https://www.welfare.ie/en/Pages/Apply-for-Certificates.aspx, although this is a snail mail version for which payment cannot be made online. To pay online, visit http://www.hse.ie/eng/services/list/1/bdm/Certificates/ instead.
So I've ordered the relevant application form, and a copy of my birth certificate. Now I need to wait for them to arrive...!
PART 3 at http://britishgenes.blogspot.co.uk/2016/07/claiming-irish-passport-part-3.html
PART 4 at http://britishgenes.blogspot.co.uk/2016/09/claiming-irish-passport-part-4-claimed.html
For details on my genealogy guide books, including A Decade of Irish Centenaries: Researching Ireland 1912-1923, Discover Scottish Church Records (2nd edition), Discover Irish Land Records and Down and Out in Scotland: Researching Ancestral Crisis, please visit http://britishgenes.blogspot.co.uk/p/my-books.html.