This is my Akali build. I use this build every single game, with the item order usually the same. I've tried several different builds but I find that this one works the best.

The runes I have here are fairly simple though they may cause some confusion.
The AD runes/quints allow me to have double passive at level 1.
The HP/level are just my personal preference, they can be swapped out for whatever you'd prefer.
The flat AP glyphs/quints are required to get one of your passives going at level 1 without any AP items.

I run her with standard caster masteries. I feel that the reduced time on summoner spells and extra reduced cool downs is much more important than being able to get both passives without items. This allows your early burst to be effective and you spend less time waiting and more time mashing buttons and killing people.

Hextech Revolver: This nifty item gives you a significant boost to your spell vamp along with AP, it helps you go toe to toe with tougher characters and come out on top.

Mercury Treads: Usually I just get boots 1 unless I'm getting harassed pretty hard. With your masteries they give enough movement to hold off on upgrading unless you need the passive that mercs gives. While I think that CC reduction is key on Akali since her jump is not like Katarina's and is only usable on enemies, if their team doesn't have much CC or you happen to be stomping them, then Sorc shoes are a very viable choice

Rylai's Crystal Scepter: This item gives Akali some helpful HP and a much needed permanent slow. While the Gunblade can slow for a short period of time Rylai's gives you a reliable permanent slow. This item should be finished before the Gunblade to make chasing easier.

Hextech Gunblade: Here's the main focus of my build, this item gives AD, AP, Spell Vamp, and Lifesteal, the stats on this item boost both of her passives to a substantial amount also. As a melee assassin with AP focused abilities I don't see how anyone would not want this. This will make your second portion of Q hit harder, you will be gaining life from everything you do to your enemy, and to top it all off, it has a nice passive for when they try to run away at low HP.

Lichbane: This item is amazing. Yeah, it has mana on it, and yeah by the time you get it you're only going to have 300 or so AP which is about half the amount most AP carries have by that time, but here's the thing, you're also toting around 160 AD, and when you R-Q-E someone, you've got an auto attack in there hitting for around 460 AD if not more depending on blue pots or whatever. It has the added bonus of giving move speed and more MR, which nets you around 100 MR (50% redux) which always helps.

Void Staff: Normally my games never get this far. Either the enemy surrendered because I just rolled through and quadra killed them or I've caused enough destruction that my team mates have pushed into their base, but if for some reason you have either mega farm going on or you can't seem to get anywhere, you should pick up this item. As I said before, you have no MPen except for your masteries and this will help with that. While this build does give Akali more versatility by being slightly hybrid, most of her damage IS still from AP, so you need that MPen.

Rabbadon's Deathcap: This thing is only really here as a theoretical item, I've never gotten this far, and I doubt anyone will. It's mega expensive, but really, why NOT get more AP when you can?

Your Q is obviously your bread and butter. It's what you harass with, please don't last hit with it unless you have to, while energy does regen fairly quickly you CAN starve yourself of energy in a time you'd need it the most.

Your E is your second most important skill, it's what you get most of your healing from your spell vamp off of. Get yourself in a crowd of creeps and hit E and you'll get some nice health back. It also of course is a nice follow up to your Q just for more damage output.

Your W skill is amazing. I've seen lots of guides and heard lots of people say wait till 7 to get this skill. That's stupid. Your W can get you out of plenty of ganks from the enemy jungler just on account of the fact that they can't see you unless you attack. That being said though, only 1 point at level 4 is needed until you max Q and W.

Finally we get to Akali's ultimate. Her ultimate is fairly neat. It won't actually do as much damage as her Q does, but what it lacks up for in damage it gives in mobility. You can use it to jump to an enemy champion from behind a wall, to start a gank if you have Rylai's, or even to escape if you've overextended and the enemy creep way is behind you. This is an amazing skill, and really, when used in a crowded team fight where the enemy is dying, Akali reminds me of a pinball bouncing around when using this.

Flash: Flash is great, You don't really need ghost as ghost is mostly for chasing down opponents and who needs that when you can jump to them and slow them? Flash can get you out of hairy situations before level 6 and after you've eaten up all 3 of your charges.

Ignite: Ignite is a great spell for early kills or even to have up in case someone like Mundo, Yi, Sion, or anyone with a large amount of lifesteal/regen. It doesn't last long, but when your whole team is focusing on them, well, they're not going to fare too well with ignite up.