Akali is one of the greatest and i dare to say op champs in League Of Legends.Why do say this?well if you come to think of it no other champion has such a big damage output and survivability from the begining of the game.Overall Akali is able to get into a fight with several champions at a time,pick a kill and still be able to escape.
Bellow you'll find a build which hopefuly it will help you play her decently.

For runes on Akali i use either Arp Marks or Magic pen Marks,HP per lvl seals ,flat Ap glyphs and flat Ap Quintessences.

For masteries i actualy have 2 builds depending on the mood i have:P
For a more ofensive build on Akali i use a 22/0/8 build avoiding marking any points in the crit option.
For a defensive gameplay on Akali i use 9/21/0

First item you should get is Doraa's Blade which gives you a nice bonus dmg a small amount of lifesteal and 100 hp.The next item you should get are boots you'll be trying to go for Boots of swiftness but depending on the team they'll have you can make either Swift Boots or Mercury treads for more survivability.
One core item in Akali's build in my oppinion is Hextech Gunblade which you'll start building it from Hextech Revolver ar first and Bilgewater Cutlass after.Once you have your gunblade you'll be starting to go for Rylai's Crystal Scepter,i personaly start with the Giant Belt but you can start building it starting with Blasting Wand or Amplyfing Tome depending on the needs or/and gold.Imediatly after these items you can continue with either Rabadon's Deathcap or Lichbane,i'd suggest you start with Rabadon's Deathcap for more ap meaning more dmg.
After doing these items you can either go for some more Ap or for more Surv items,depending on the needs or mood.

I start playing Akali with Mark Of The Assassin maxing this skill first followed by Crescent Slash and Twilight Shroud taking a point in her ultimate at 6/11/16.

Depending on your gameplay i suggest you either take Ignite and Flash as summoner spells or Exhaust and Flash.Once again,chose your spells depending on you gameplay this is just a suggestion.