Explaination of Skills:
Twin Disciplines(Innate):
Discipline of Might - Upon obtaining 20 ability power, Akali's hits deal 10% bonus magic damage, increasing by 1% for every 5 ability power gained thereafter.
Discipline of Force - Upon obtaining 10 attack damage, Akali gains 10% spell vamp, increasing by an additional 1% for every 20 attack damage gained thereafter.

This is what immediately makes akali more reliant on both AD and AP, this is simply why you can melee people for hard damage with a huge amount of AP. And this is what makes you able to dodge Life steal items.

Mark of the Assassin:
Akali deals magic damage and marks her target for 6 seconds.
If she hits a marked target, the mark triggers dealing the same magic damage and restoring energy.

This is the spell which you are gonna cause the highest amount of damage with. You use this as your first ability as you start a fight, it works for last hitting minions too if needed.

Twilight Shroud:
Akali throws down a cover of smoke for a few seconds, becoming stealthed while in the area. Enemies in the smoke have 40% reduced movement speed and reduced attack speed.

You can use this spell to make sure people cant run, they will be so slow you can close in on them and finish them off. Its your best way of escaping aswell, as used in jungle can confuse your chaser. In certain scenarios you will have use of the armor/magic resistance to survive a big hit, such as a nunu firing absolute zero and you attack them, the armor/resistance will make you take the hit better(if nunu actually survives long enough to charge fully).

Crescent Slash:
Akali flourishes her Kamas, hitting nearby units for physical damage plus 30% Ability Power + 60% Attack Damage.

This spell doesnt hit as hard as many of your others, but you should use it often, as it activates trinity force(which means you hit like a truck with your next attack), and you can farm EASILY with it.

Shadow Dance:
Akali moves through shadows to quickly strike her target, dealing damage and consuming an Essence of Shadow. Akali gains a new Essence of Shadow every number of seconds, up to 3 total.

This is what makes you one of the fastest champions in the game, 1 essence is the whole cost, about 1 second cooldown(at rank 3), AND damage, not to forget haves a longer range than Shunpo(used by katarina). you can even end up using this spell three times in one fight, because you can use it straight after Mark of the Assassin to close in and hit them for bonus damage, you might also just fire it to activate trinity force(not likely but maybe). You can also escape by using it in the jungle or on far away minions!