I would recommend looking on each of the different airline�s websites and look on Wikipedia to see what airlines serve the respective airports that you wish to fly from, go to or connect at. It can be a bit of a boring process but it will get you the cheapest deal. There are no direct flights from the UK or ROI to Vietnam as I�m sure you know and I would expect you would travel into Ho Chi Minh City correct if I'm wrong on any of these.
From Belfast you would have to take a flight to the UK mainland probably London or perhaps Paris for a direct flight to Vietnam (with a probable stopover in Bangkok) If you decided to go through London you would have to take another connecting flight onto Vietnam. Alternatively from Dublin you can take Air France via Paris or Lufthansa via Frankfurt which both serve Ho Chi Minh City. If you know which route to take to get there then tell me and I will give you my opinion about it if you want. I would personally recommend going through Paris and taking Air France or Vietnam Airlines onto Ho Chi Minh City.