Hello and welcome to my hybrid build of Akali.
First of all i want to say that i play her depending on my team. If we have enough AP in our team i play her this way, otherwise i play her with Lichbane, Soulstealer and Zhonya's. But as there are enough very good AP builds on Mobafire, i decided to show you my hybrid way of playing my favorite champion.

Runes and Masteries:
With the Runes i want to unlock both parts of Akali's great passive, thats why i took Flat AP to get one and 2 Points in Brute Force( combined with Dorans Blade) to get the other one.
Furthermore i took Dodge Seals just for more defense and Armor Pen Marks( it really doesn't matter if you take armor pen or magic pen cause you go hybrid).

Summoner Spells:
I take Exhaust and Flash. Flash explains itself just for getting away in trouble. Now Exhaust. I see many Akalis taking Ghost of Ignite as second Spell. Sure it's not bad but you don't need ignite cause you doing enough damage normally to get you target down and ghost is not really needed cause you have flash and Twilight Shroud. But Exhaust is just great for her in every 1v1, especially after the Patch on January 4th, cause it nearly guaranties you a kill or an escape.

Items:

Start with Doran's Blade and a Health Potion just to get your second Discipline up. I take Doran's Blade here instead of Longsword just for more surviveability.
Grab your Sheen ASAP cause you will gain a lot of more damage when you trigger the Damage of your Mark of the Assassin with an Autoattack.
Then comes the Question of boots. When i build her The AP way i use Sorcerer Shoes but in this case I suggest Boots of Swiftness or of Mobility cause you need the speed to chase and flee. Go on with Phage to get your Trinity Force ready ASAP. After that i like The Guinsoo's Rageblade, wich is just great for her passive and for pushing towers as well.
Then you have to look how the match is going on. If you get focused or have problems with chasing, then buy Rylai's first. If the enemies make the mistake and don't focus you first(this is very often the case cause they normaly focus a normal caster like Ryze or Annie and the they underestimate your damage untill yopu tear them apart) then buy the Gunblade first for just more burstdamage and lifesteal.
My last item is Banshee's Veil for just more defense against crowd control.

Champions Laning with and not and possible enemies for mid:

If you are going duo lane: - any heavy CC like stun or taunt is just great(Rammus, Shen, Sion, Evelynn....)
- Don't go with someone very squishy who is relying on heavily farming minions, cause YOU NEED THEM and you don't make kills on your lane with such lane partners and YOU NEED KILLS TOO( e.g. Cassiopeia, Katarina, Nidalee ect...)

If you are going mid:
The chance that you are getting feeded and totally rule the game is very high if you mid-opponent is an Ashe, Kartus, Tristana, Lux, Morgana ect...( everyone who has low life and needs a bit for more damage)

Difficult, but possible are champs as Vladimir, Malzahar, Annie, Mordekaiser

Very hard and nearly impossible(ask your team for ganks all the time!) are e.g. Mundo, Nunu, Cassiopeia
It's so hard cause they either harras you downextremely fast or they just don't die AND deal heavy damage. Sure you can harras too but you are very relying on getting lvl 6 and till there it is a long way against such champs.

I hope that this build could help you playing Akali as true Hybrid