A very basic and straight on guide to Akali.
I haven't been playing her for long but I also come to understand how champions are played very quickly. I have also seen several Akali's being played on enemy teams and my curiosity just couldn't be held back, I had to study and play her.
And here is my conclusion:

Play her as a carry. Don't initiate like a tank. Run when you need. Have tons of fun.

Runes -> Comparing defense to damage, runes give more damage than defense when a book is FULLY dedicating to one option. Straight AP gives more damage than having defense runes could make up for, or even magic penetration runes for that matter.

Masteries -> The idea of Akali being "tanky" is to improve her survivability without crippling her damage. So seeing her mainly as a carry, one would want to max out her damage, and this means giving her as many boosts from as many sources as possible before getting into the game. Now, since she's an AP carry (in this case) shes going to need her 21/0/9 setup. The offensive tree for magic damage and the utility tree to chase and get back into fights or defend the base ASAP. (If the bug isn't fixed with the site yet, then you want to fill up all the masteries in the third column from the left of the Offense Tree, then 3 points to Havoc, and one in Executioner; Utilities would be maxed Good Hands and Swiftness, with one point in Transmutation and one in Improved Recall)

Summoner Spells -> I really only ever use Heal and Ignite on all my champions and builds. It's just always worked for me; being able to survive a little longer or get out of a fight with less worry, or just guaranteeing a kill when my skill rotation burst isn't enough. The way I see it: I prefer Heal/Ignite combo, but go with what your comfortable with. Remember, this Akali is played like a mage.

Items -> She gets all she needs. Doran's Ring initially grants her passive, then Frozen Mallet procs the other passive. Frozen Mallet and Rylais give massive early game life making Akali very hard to kill. Void Staff and Abyssal Scepter for magic pen. and magic resist reduction. Rabadon's Deathcap because... well... comon, it's a massive AP boost that every mage ought to have. Your boots are really based on preference or situation: Sorc Shoes for extra carry ability, Ninja Tabi for your armor and damage reduction, or Mercury's Treads for that tenacity and magic resist. (Although Ninja Tabi is where your armor comes from for tanking)

The way I see it: have a dedicated rune book for as many general purposes as possible.
So, since I simply don't have money to spend on the game or the time to play enough to earn massive IP, I only have 2 types of rune books: Straight AP and a Crit. Book which is still being worked on.
The point of giving Akali straight AP runes is to give her that fantastic 83 extra ability power that will just make her more of a carry while being able to sacrifice an item or two to be tanky. (which she really isn't even sacrificing due to everything her needing being provided for)
So: Runes are as a primary source of damage.

21/0/9 AP set up

Max out the AP column in the offense tree, max out havoc, and top it off with executioner.
Max out good hands and swiftness in the utility tree and put a point in improved recall and transmutation.

The utility is important, it'll allow you to move faster than most (if not all) enemy champions. Grants a tad more spellvamp, and of course lets you get back into fights faster if you die.

The listed item purchase is for straight AP carrying. Although a more viable build might replace an item (not deathcap, hextech GB or boots) for Will of the Ancients. This so that you have higher base AP and damage and massive spellvamp. The trade-out though reduces your magic resist and lowers the chance to reduce a target's resists to the point where Akali's damage might actually be higher than with higher base AP. If you find you are chasing too often and for too long then you can also go for Boots of Swiftness.
With this build, you have 724 AP without Baron or potions, 34% spellvamp, 2475 life and 51% damage reduction with a nice 151 AD. It's pretty viable and nukes the hell outta enemies, even if building tankish.

To build a more tankish and survivalist Akali, here are some items to consider:
Zhonya's Hourglass (keeps AP high and some Armor plus the active), Spirit Visage (use with high AP build and WotA + Hextech GB to maximize spellvamp; also some Magic res. and life), Randuin's Omen (lots of armor, nice life+regen, and a chance at auto-countering enemies)

Personally, I like the carry build.

I find it best to initiate with your Q in any fight before actually going in for kills. This way if your target get surrounded by his teammates you don't go in committing suicide, and if you need to hold back then you can just harass for lulz.
If there's an easy opening then after throwing your Q go ahead and ult their face. A nice 500 odd dmg nuke to the face oughta scare em, it also gets you in range to activate your Q's mark dealing some nice extra dmg (which will always be more because it also counts as a basic attack, so Q dmg plus 151 AD), if your prey stays in range then go ahead and E them, it's not much dmg but you gotta keep putting the hurt on 'em. After your rotation if the target isn't dead then go ahead and pop the rotation again, and continue doing so until you kill 'em. If they're getting away and might get outta range of your ult then ignite (or if your ult is on the long cooldown.)
I like to use Akali's W when the other team is chasing your own and you all need to get away, its not much of a slow, and not for very long, but it's at least some support for your teammates to survive a fight. Also use it for it's intended purpose (hiding in the middle of a fight to either get your skills off cooldown or wait for an opening to run away.)
And of course, heal whenever you need to. Just don't forget about it. It's a lifesaver.