First off, I am not a super pro player, I really am not, but I was thinking about new ways to play Akali, and went looking for a AD Akali guide. There was none, I started playing around in the guide creator testing different builds, and I ended up with this one. I played a practice game vs AI because I didn't want to bring my build to the public yet(afraid I would screw it up). It was amazing, I mean, AI are usually easy, but it was just destruction. By the time I had full build (40ish minutes) I could 3 hit Caitlin. So here is my build, and less than detailed guide to my choices.

Pros:
Hella damage output late game
Unexpected build
Amazing farming capabilities
Snowballs like crazy
You are a ninja
Cons
Squishy early game
You will get focused like crazy
Expensive build(need them creep kills)

Greater Quintessence of Scaling Attack Damage - If you can't figure it out, its for the AD.

Greater Mark of Critical Damage - You are going to have 100% crits at full build, this extra crit damage does alot.

Greater Seal of Health - Akali is quite squishy early game, without crazy burst from AP to make you scary, you need the extra health.

Greater Glyph of Scaling Cooldown Reduction - Faster cooldowns = more Q+E combos = more damage. This is good

These are just suggestions, it really comes down to personal preference, but I highly recommend the flat health runes for early game.

These are my favorite Masteries, with 9 points in defense so Akali isn't as squishy early game.

--Super important Items--

Avarice Blade - You need the gold for these rather expensive build.

Trinity Force Akali is pretty squishy till we get this, you really start getting going here.

Phantom Dancer - Easily one of the best items for an AD carry, you don't want to trade this out.

Atma's Impaler - The armor and crit chance and crit damage are all just too good to give up, get this item.

--Not so necessary items.--

Infinity Edge - This is such a good item, you can build with other items, but really, why would you want to?

Youmuu's Ghostblade - More crits, more crit damage, team buff, what more could you want?

AD Akali can certainly work with other items, but I find these to be the bet setup.

This is still a work in progress guide, I will add more in-depth answers, along with skill explanations, Summoner spells, and tips and tricks for this build. If you have anything that you think should be in this guide, feel free to drop it in the comments, I could use all the help I can get.