Hey this is Zemperfy. I'm an avid Akali Player. And if by some reason you decide to look me up, know that this is my second account. As I didn't like the name on my previous account. I've used alot of builds taken them and changed some things or tried my own. I use this build often and I'd say i usually go around 16-2 with her a match (assuming that there is no lee sin. As he can pull me out of shroud easy.
With this I'm going to show 2 different builds.
One Is going to show you my Akali Build with Snowball Items, My second is what you should Build if you can't get stacks or you feel snowball items are not needed. (Even though from my experience Snowball Items are one of the best paths for Akali.

I take Flash and Ignite as my summoner spells. I like Flash for the ability to drop shroud and flash away. (Over a wall, Into a bush ETC.) And Ignite is to help with my second passive and first blood.

My runes are set up to give me my passives and durability.
Greater Mark of Attack Damage Gives you Early spellvamp. Must have over the second passive.
Greater Seal of Vitality Give you Hp per level seeing as by level 6 they are better then the flat Hp seals. And you have quints for some early game HP.
Greater Glyph of Ability Power Gives you only about 9.91 Ap, yes I know not enough for passive, But after you Ignite someone it gives it to you, seeing as the passive give you only about 10 damage early game you really only need it for attacking champs because your Mark of the Assassin On hit effect should allow you to last hit minions with ease.
An alternative if you choose NOT to take Ignite or feel that using ignite just for passive is a waste is to Get flat AP quints, 5 Flat Ap glyphs and 4 Ap per lvl Glyphs for lvl 1 passive on both.
The build I use is 14/16/0. I put 16 into defense to get the extra Health and to be a little tougher early game. 14 in offense for the obvious spellpen, a little armor pen and the 2 points for 2 AD to set off my passive.

First is the Main part of the build
Boots of Swiftness - Has the desired movement speed for chasing/escaping.
Trinity Force - Health, Movement speed, Sheen effect, and slow. Nothing much more you would want from an item. (You can get Lichbane depending on what secondary items you want. But I play with trinity almost always.
Sword of the Occult - OMZG STACKING ITEMS R NEWB! No. With Akali you will kill people. All the time.
Mejai's Soulstealer - Same concept as Sword of the occult.
Now are Optional items.
Guinsoo's Rageblade - Staple hybrid Item, AD/AP/AS Amazing and favorite item.
Rylai's Crystal Scepter - Slow/Hp/AP. The reason why I don't build this is because I personally don't feel like you Need the slow. And seeing as I build hybrid and not ap AND The Trinity Force is Good for slowing. But if you took Lichbane Get this for sure.
Hextech Gunblade - Another amazing hybrid Item, Lifesteal spellvamp On use slow. All around good.
Force of Nature - Move speed, Magic resist, Health regen. Lovely item for heavy Ap Enemies, and for pesky Teemo shrooms. Also helps you mob around for kills or healing while farming.

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SECTION 2

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For those people who dislike snowball items or just suck too much to get kills.
Boots of Swiftness
Guinsoo's Rageblade
Trinity Force OR Lichbane Depending on what you want for your optional item(s)
Hextech Gunblade
Optional items are
Rylai's Crystal Scepter (If you took Lichbane )
Rabadon's Deathcap If you took Lichbane
Its really just preference after main build...

Shadow Dance > Mark of the Assassin > Twilight Shroud > Crescent Slash
You want Mark of the Assassin As often as possible. This gives you burst damage. Twilight Shroud Gives you a slow and a stealth. Unlike alot of Akali I max this when I can't Level Shadow Dance or Mark of the Assassin for the slow. Crescent Slash Is just way not worth it. It doesn't do much to minions till late game, And you shouldn't use it against champions as it just drains energy and does stupid little damage.

Pros
Amazing Damage.
Great Chaser/Escape.
Powerful in 1 vs 1 against carrys and team fights.
Strong Farmer Mid game.
Spellvamp/Lifesteal makes it easier to lane. (Or kill creeps in jungle for a quick heal when you cant go back)

Cons
Squishy if not played right.
Oracles own you.
Ad carrys are scary if you aren't the one initiating the fight.
AP nukes are also scary. So jump right in and kill them like nothing.

Pick off carrys then go for support. Focus on squishies. Twilight shroud has more then just stealth to it. Use it to slow to stop opponents from Either escaping or chasing. Remember. You do NOT start fights, You finish them.

Please please please please don't just make ****ty comments, add constructive feedback. But not stuff like "Ap Akali is better." Because well. The point of this is HYBRID akali not AP.