Akali is a burst damage assassin that can be built ap or hybrid AD/AP. This guide explains how to make a hybrid akali[Majority AP] with op burst and unfair lifesteal.

Pros
Akali can burst people down quick
Akali has good escape ability
Great solo laner
Akali is one of the best if not THE best at chasing down fleaing enemies

Cons
Akali is squishy
CC is a killer
Oracles and Silences make your escape difficult

Akali wont dominate until mid to late game. Early focus on using your q to last hit minions, and your W to check bushes. If you have to get your CS off turrets, do it, you'll get your chance to push back later. If solo laning, if you don't buy wards[which i would highly recommend] watch your enemy habbits. If you have more health than your enemy and they engage anyway, be weary of a gank.[This shouldnt be a problem because you shouldn't be overextending too much anyway]

For my runes I chose:
9 AD/Level Marks
9 HP/Level Seals
9 AP/Level Glyphs
3 Flat AP Quints

After trying out different runes I decided I like this one the best. The AD and AP per level naturally sets off her passive at around level 10. HP is taken where you can get it as Akali due to her low health pool.

*You could change AD with armor pen or AP with magic pen
*You could also buy Flat AD Quints to set up her naturally spell vamp earlier, making her a tougher laning presence.

I choose a 21/0/9 Mastery page.
The Offensive tree with help max out your burst as much as possible, and the utility tree will help you get to your prime.

*Take defense over utility if you find it more helpful, I just prefer utility

1)Boots of Speed
2)Hextech Gunblade
3)Rylai's Crystal Scepter
4)Rabadon's Deathcap
5)Lich Bane
6)[Personal Preference]

This is the item build I prefer.
I like to get Boots of Swiftness with most assassin champions, I feel it helps get the gank right, and will bail you out in a rundown.
Hextech is Akali's most important item, even after the recent nerf to it, once you get it you will enjoy nice burst and satisfying lifesteal.
Rylai's and Rabadon's are my next items, the Rylai's gives you some survivability as well as a slow for when your enemy starts to run. Rabadon's is important to get before Lichbane to ensure you have quality AP before you go in that direction.
Lichbane will finalize your dominance of the battlefield adding your ap to your attack after a ability.
The last slot I leave open for preference. Depending if you even get this far you could build anything that fits your current situation whether it be surviability or more damage.

Simply given your combo should look something like this:
Q champion you wish to die, R at them and then melee to activate your Q, E right after melee.
Your W will be to hide when being targeted or to help allies being chased by slowing enemies. Once you have Lichbane and you want to burst down an enemy even with your Q on cd you can R at them and melee them to activate Lichbane [The burst wont nearly be as good but in certain situations it helps]

Q-1,3,5,7,9
W-2,14,15,17,18
E-4,8,10,12,13
R-6,11,16

Pretty basic: Max out Q as fast as possible, followed by E, then W. Like pretty much all champions, your ultimate when possible.

Ignite
Flash

I choose ignite and flash for summoner spells, I like ignite to finish off enemies [and for Tryn]
I find flash more helpful than ghost. Offensively you only need that quick chunk of distance to get into R range. Defensively I like it better to flash through walls [and especially Jarvan's wall]

Akali's potential is great, and you will find yourself killing squishys in seconds, and picking off enemies one by one. The key to a good Akali is picking your fights and knowing your surroundings [For ganks and possible escape routes]