Hi Dexter,
You haven't given me much to go on here, so my answer can only be that depends on what you actually know. You said that you know his name and where he lived. But you actually would know more than just that. You have to use the clues provided by his life to help you. I will just share with you what I'd do. So sorry if this turns out to be a very long answer.
So first things first, I'd narrowed down the period in which he died by asking the usual questions, such as: Did he die before you were born or after? If before, then was it before the birth of his last child or after? What was the last major family or historical event at which he was known to be alive? Did he die before or after your grandmother? Who inherited whatever property, if any, when he died? I'd also find out all the names he was known by. This part could be easy enough if you knew him or have relatives whom you could ask.
Once that's done, I'd start a search for the records he left behind in the family first. Believe you me, someone usually has something. Does anyone in your family have any of his papers? A funeral card? The funeral service bulletin? A death certificate? Even if you aren't sure, you need to ask yourself whom is it likely to be? Who took care of making his funeral arrangements? What funeral home did they use? Where was he buried - was it in a family plot? Where was it and who has the deed? At this point, I'd review my findings. If I still hadn't gotten the exact date of death and/or burial yet, I would press on by looking for church records and public sources of information.
If I narrowed down the period, the first thing I'd do is a Google search for his name and the Trinidad and Tobago Gazette. A few of them are online. I'd look through the section for grants of letters of probate and administration. These will give you his date of death and his last address. Whether or not I got a hit, I'd go to the National Archives of Trinidad and Tobago {NATT) to search the newspapers. If you got a date, you'd want to look for an obituary to make sure you have the right person. If you didn't, you'd want to look for both an obituary and for the public notices for letters of Administration and Grants of probate. If he died in an unusual way, it might have even made the paper.
If I still hadn't got a date, but I got an obituary, the next two places on my agenda would be the church and the cemetery mentioned in it and I would see what I could find out there. But before you go to the cemetery check with the National Archives to find out whether it has the burial/cemetery book for that particular cemetery.
Just an aside because I'm presuming that you live abroad and may or may not be familiar with T&T obits. Some obits may give you the date the person died and their age. Some don't. Most will tell you where the service was held. And I always hope that it would not say the house of mourning - because I may not know where that was. But if it does, they may hint at the denomination by saying according to "whatever" rites.
Hopefully, by now I would have gotten a date and am certain that the person of interest is my person of interest. I would have also gotten a locality where the person may have died. This is when I'd go to the Registrar's Office and apply for a search and/or a death certificate to find out the cause of death. Depending on the year of death, you may get the certificate that same day or you may have to wait up to 6-8 weeks to get the result of the search and then you can apply for the certificate.
This is why all of the previous steps are necessary because when you are sure of your dates, it shortens the waiting time and minimizes the expense. One last thing, if you don't live in T&T and are making a research trip for this purpose, my suggestion to you is to go for at least four weeks and travel in a month when there are few or no public holidays. Come in on the Sunday or the public holiday if you think you'd need the day to get yourself sorted but take the red-eye in if you want to hit the ground running and maximize your time...because all this will take time. Especially the church part because only a few of them are open Mon-Fri. Most have 2 office days a week. Hope this helps...