The only way you can bring your dog to Hong Kong is as cargo, unless you have a very very good reason for an exception.
Last year we brought our two dogs from the States using Continental. They took a flight to NY for a layover (they were let out and walked) and then came staight to HK. They did good, but I would not want to leave them at HK cargo depot for too long. The place is noisey. Continental and Cathy Pacific are both highly recommended. Reserve your space ahead of time.
Hong Kong rules are pretty strict so do read all the instructions, print them out so you can refer to them later (including the maps), and be prepared for a lot of running around once you get to the HK airport. Follow the recommendations listed and your dog should be okay.
If you have the time and patients you can do everything yourself and save some money. Put together a check list and plan everything out.