I've been playing Akali for a long time and this is my first guide. Constructive criticism is welcome.

Doran's blade as first item to trigger your passive, bonus heath for survivability
On your first trip back you should be able to get at least an amplifying tome, If you and your team are doing very well replace Hextech Revolver with a Mejais and Rylais Crystal Scepter with Hextech Revolver. Get some Elixirs of fortitude every now and them when expecting a team fight or w/e. If you're not tanky enough or dying too much replace Rabadon's Deathcap with a Zhonya's Hourglass for armor, ap, and a lolz active.
For extra damage replace Mercury Treads with Sorcerer's Shoes

Best: Flash, Ignite
Good: Exhaust, Ghost
Acceptable: Teleport, Cleanse
Bad: Smite, Clarity, Fortify, Rally
IWillGoToYourHouseAndKillYouIfYouPickThis: Revive
Situational but not recommended: Clairvoyance

Q: This + your normal attack is your main damage output
W: Very good for escapes, slows, stalling, I guess a bit of armor too. Use with Flash for escapes
E: This is meant to be your last resort and if their HP is SO low and everything else is on cool down because is very short range, energy consuming, and not very useful.
R: ******icly good chaser very useful and the main reason why running from Akali is futile.