I can only speak for HK (and to a lesser extent, my home jurisdiction of Singapore), but I would say HK is pretty good. A lot of the international law firms have offices here, pay is very decent (I had a quick look at rollonfriday - it appears to be slightly above the London office of my firm for trainees + HK has far lower tax rates), and the TC itself is very similar to what you'd get in the UK (2 years with rotations and secondment opportunities). As I understand it, firms are generally committed to keeping their trainees because the cohorts are much smaller, so retention rates are extremely high. However, most of the international law firms in HK focus on Corporate, Finance and Disputes, so more niche seats like Employment and Tax won't be available unless you choose to go to a local or specialist firm.
International law firms in HK recruit predominantly through vacation schemes (eg
http://www.allenovery.com/careers/hong-kong/en-gb/students-and-graduates/Pages/default.aspx), which run from June to August. Most last a month, although some last 2-3 weeks (eg Slaughter & May). The same timeline applies - offers are given two years in advance, so you probably should be looking to apply this year (ie now) for summer vacation schemes in 2016 with a bid to securing a 2018 Training Contract, unless you want to take an additional year out.
I should also add that fluency in Mandarin Chinese is very important; I have heard of people being taken on without it, but you're at a definite disadvantage (although in your case, you might be able to substitute with Japanese/ Korean if the firm does that sort of work out of the HK office). Fluency in Cantonese would be useful socially, but isn't necessary for work purposes.
Once you secure an offer, you need to take the Conversion Examinations (
http://www.pcea.com.hk/); there are two sittings each year in January and June. You must finish all your Conversion Exams before you'll be allowed to enrol on the PCLL (HK's LPC), which lasts a year.
I'll be working for a UK MC firm. For local firms, I imagine fluency in Cantonese is important (which I lack).