Akali as we all know is a lethal champion to play with, as this build I have been using for quite a long time will enable you to ASSASSINATE your enemies before they even blink, unless they are tanky.. its gonna take a few hits to put them to the ground.

One thing I can guarantee you is survive till you are level 6. then you will get lost counting kills. Enough with the useless talk and lets explain few things.

Here is my combo, of course it might change according to the situation but here you go..
Always land Q before every attack, it makes your next basic attack more painful specially if you have your lich bane. you go like Q > R > Auto attack > E > Q > R

Keep an eye on your ult cooldown because you need it twice at least in case of they flash away.

Use your W wisely, it grants you vision so you can use it on walls to jump to the jungle camps if you get in trouble.

Never focus tanks in teamfights, you might run out of Ultimate attacks which will a short while to cooldown. Instead focus ADC's and assassins as they do too much damage to your team and they are too squishy.

Use your Hextech gunblade active ability when chasing down enemies, it has longer range than your ultimate so it will do just the right job to decrease the distance for you to eliminate them.

Farming might be a bit of a hassle early game and I personally don't risk getting too close from my enemy unless I can kill them.
Use your Q to farm early game and sometimes even W to cover you while you go them.. depending on your stats sync your Q and you will know the right time to throw your Mark of the assassin.

In the end, Akali is always fun to play and doesn't need much of skill. practice it few times and you'll learn to play her in no time.

This is a really short and brief guide, but I guess I covered all that you need to know. anyhow feel free to ask or correct anything I might got wrong.

Enjoy !