Stretching, stretching, and more boring stretching.
At the sme time, you can watch MMA fights or olympic sports or hentai or whatever you dig, though!
Seriously, extended stretches, all kinds of slowly progressing exercises are the way to go.
Dynamic str., occasionally ballistic s., isometric resistance followed by various relaxation drills, fifteen+ minute ass-boring s., various assisted stretching, slowly placing a high kick on a partner’s shoulder without body english, raising you leg an inch (various angles) from a high resting position for 2-3 seconds…
Know your goals. Set realistic ones. Reaching the level of that kick-chic in one training period isn’t. At all.
Achieving both kinds of splits is one, for example, if you can’t do it already.
Know how to stretch the whole hip apparatus, all involved muscles, or you’ll suffer from shortened ligaments.
You need decent core and leg supporting muscles. I take you know this already and this is probably the easiest part. V sits (and L sits for starters) are the obious pick for this one. Especially if your leg shakes and burns like you described.
Progress steadily. Don’t go for too much pain, volume is far more important here then intensity.
I’d say, five times a week is a must for improving flexibility. But do it when you are sure you can be fairly relaxed. Sometimes it’s great right after regular MA/GPP/ weights training, sometimes it can be better to do in the late evening, it all depends.
One last word: Gals have an advantage for doing these kind of acrobacy, it doesn’t mean we guys can’t but if you are determined to achieve this you have a LONG way before you.