Build is updated, guide is not. Give me a day.

The use of Kael'thas' abilties is very straightforward. The most important thing is positioning and knowing which one to empower at a time. Hint: It's probably Chain Bomb.

Almost never use Verdant Spheres on Gravity Lapse. Because of this, never walk around with Verdant Spheres. If you do and need an instant stun, oops, you just wasted your trait. The only time I personally use a traited Gravity Lapse is early game. Say you get ganked by Zeratul at level 2. If you weren't expecting it, you're dead unless you can hit both Zeratul and the enemy laner in a Gravity Lapse. At this point it the game you do not do enough damage, so any traited damage is useless. If you have a teammate with you, go for the damage, though. Lategame, damage above all else.

Until you get Chain Bomb use empowered Flamestrike for waveclear. You may think differently, but trust me, it works much better.

Both your Ults are very low cooldown. Don't be afraid to use them.

Positioning, positioning, positioning. 99% of the time, if you make a mistake on Kael'thas, it will be in positioning.

Even if you fancy yourself a fire mage, at a dinner table, NEVER light the napkins on fire. No one else will think it is hilarious as you do.

He didn't stay dead long, did he?

Mana Addict can be taken at level 1. I preferred Fel Infusion before the patch, and even more now due to Kael requiring less mana because you will now use Verdant Spheres on Living Bomb instead of Flamestrike. Even if you take Arcane Barrier, I prefer Fel Infusion simply because most of your shield will be wasted anyway. If it isn't, your positioning is the reason you're dying.
Any talent can be taken at 4. They're all rather weak. I recommend Nether Wind. Gathering Power is also worth considering vs Murky or Vikings.
Fission Bomb is a must take.
Pyroblast is now acceptable to use in place of Phoenix with its recent range increase. The choice is very situational now and depends on how hard it will be for you to get the cast time off.
Chain Bomb is a must take.
Arcane Barrier is an great talent at 16 if you are being dove. If the enemy cannot jump right on you, Backdraft is preferable as it will allow easy kiting.
Bolt of the Storm is a must take.