Passive: Twin Disciplines
It's best to have both passives as early as level 2 so you can focus on core items rather than having to go a different direction just to cope with it (follow my rune page and masteries page). Healing passive and damage passive is crucial for Akali in laning phase, both survivability and damage output.
Q: Mark of the Assassin
This skill is your primary damage ability, alongside with your R ability. It's focused on your ability power and does an insane amount of burst. The mistake that most people make is using Q and then instantly using R, arriving at your opponent too early resulting in a wasted auto attack and a late Q trigger.
The cooldown on this move is short. And the duration of the mark on the enemy is higher than its cooldown by a few seconds. So to surprise your opponent and fully use your damage output, do this: Mark your target, wait for the cooldown to be finished, use your ultimate, trigger Mark, Mark again, trigger second Mark. You bascially trigger your Q move twice under a second.
Great alongside with Rylai's Crystl Scepter both for catching a fleeing enemy or if you are fleeing away yourself.
W: Twilight Shroud
It's good for almost all situations. Just be careful, when you use it carelessly and get caught with your pants down because your shroud is on cooldown, you're forced to either use your ultimate (most cases, you wouldn't ulti at their minions towards their tower) or worse case scenario, flash. Use Shroud to last hit minions if you find yourself getting stomped by your lane opponent. And if you control the lane, use Shroud to push your opponent back out of XP range. Use Shroud on the edge of brushes to highly increase your juking. If they try to find you, their going to be near your shroud and not on the very end of the brush that that Shroud was attached to.
Once again, don't keep spamming this move because you're going to eventually need it when you're getting ganked. Also, Shroud increases your MR and Armor, so sit in your shroud when Garen spins near you, a tower shoots you, or if Karthus is using his Ulti.
And lastly, while in a lane, when you use Shroud to run towards your opponent to Mark them with your Q ability, run away while you're still temporarily visible. And then the INSTANT you become invisible again, you run back towards them to get a close shot to trigger your Mark. You run away while visible in shroud to make them think that you're retreating, but when you go invisible again, run back towards them to trigger your Q. It's fun when it works out in the end.
Akali will continue to auto attack as long as she is UNCLOAKED in the shroud. However, if she becomes CLOAKED note that she will STOP auto attacking, which can mess up your combos at times (primarily your Q+hit)
E: Crescent Slash
This move is usually only good for farming. Never spam this move when you're chasing someone or in a big team fight, because before you know it, you barely did any damage with it and you just wasted all of your energy. But it is also very situational to use it to gank someone, unless you absolutely know you're going to kill that person under 5 seconds, then knock yourself out using E for that mini damage.
Keep using this move to a minimum in team fights. Your Q does 5x more damage than your E and also does not refund energy.
R: Shadow Dance
This is a great juking and magical damage ability. When used in combination of "Mark of the Assassin," make sure you wait half a second before leaping at your enemy with R, because if you leap in too soon after a Mark, your Mark wont trigger until after your second auto attack, assuming you want to attack right away.
Make sure you don't spam this move, especially early game, because the refresh timer on an additional Essence of Shadow is very long (at most 20s). Ganking with only one Essence of Shadow in most cases would be a bad idea, because you won't have a second to when they Flash/Ghost. Keep in mind you gain an additional Essence of Shadow for a Kill/Assist too. Later into the game, you can use this ability on creeps or other enemy champions to catch a wounded champion.