I had made another Akali build earlier (as you can see on the right under 'Author's Builds') however, although it works, it lacks seriousness and is more a fun build.
However I noticed recently that there lacked an off-tank AP Akali build. So I thought I would make more effort and create this build.
Enjoy

ExamplePrime

Disclaimer: This build focuses on AP so if you prefer AD or Hybrid then this isn't the build for you

Another Disclaimer: I don't use Stack Items on Akali, although with her excellent escape tools its perfectly viable on her. I don't use them however because I hate the idea and implementation of stack items. It also just isn't my playstyle, add them if you wish however.

Ok. First up I want to talk about how you build Akali
I see a lot of damage builds with Akali and I hate to burst your bubble but most of the time Akali doesn't auto-attack. Although it works well on her because of her passive its more effective (burst damage wise) to build AP on Akali. In the end you will spend more time using her abilities - or as the way I teach you in this guide you will.

Masteries

First I'm going to start with the masteries. As it is the Offense Tree is pretty useless for mages as even the 15% Magic Penetration isn't much. The Utility tree is mostly about mana regen which Akali doesn't have much of. You could go for experience boosts etc but since this Akali build goes with a lot of health I advise full defence tree and the magic penetration.
It not really worth aiming for the experience and gold upgrade as its only 300 Gold in a 45min game or so and 5% experience is virtually nothing when your level caps real quick anyway but if you want to go 0/21/9 then do so

But to me, 9/21/0 seems to be the most logical choice so with this build thats what I advise.

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Runes

As it is I can't stand Runes, just now they cost so much IP and I would rather buy new champions instead of runes so I currently run with these four runes all the time - ~ Greater Seal of Vitality ~ ~ , this is because I usually play Tank or Off-tank champions and this also has heavily played a part in the way I play every champion and this build.
If you wish to use the same runes as I do, be my guest. If you want me to clarify why I use these runes for Tanking then you can ask but I'm not going to explain here.

I will clarify my choice of the best runes (IMO) for Akali.

First off is the Greater Quintessence of Fortitude.
= 97.2 Health
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This is what I would call the default Quints, 100 extra starting HP is rarely worth passing over and I would advise it on almost all champions. These quints greatly improve your early game and laning phase and haved saved me who knows how many times. A must buy.

Next, for the Gyplyhs are the Greater Glyphs of Force
= 27.54 Ability Power
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This gives you ability power per level which is obviously good.
But surely, you say, going for Glyphs of Potency would be better as it gives you a much larger boost at the start for some early kills and maybe a first blood?
Well maybe yes, but after level 6 the Force Glyphs overtake the Potency Glyphs and to be honest, you won't be aiming for many kills until level 6 because thats when you get your ultimate is what gives Akali her real strength

For the Seals I would recommend the Greater Seals of Vitality
Greater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of VitalityGreater Seal of Vitality = 174.96 Health
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Again I would recommend these over the Seals of Fortitude as they overtake the Seals in usefulness at level 5 and the boost they give is quite substantial (175hp at level 18, add on your Health Quints for roughly 270hp. Thats an entire ability worth)
I would say these are better than the armour or dodge seals as those seals only boost your defence against physical attacks which Akali usually doesn't worry about, people usually try to stun you and lock you down with spells so more health is better as you will be aiming to have a high magic resistance anyway

Finally, the Marks should be the Greater Marks of Insight
= 8.55 Magic Pentration
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Magic pentration. Means your abilities hit harder which is good. These are chosen over the Greater Mark of Force as the extra pentration will give more benefit than these 2nd best AP marks will

The two spells I recommend are the two spells I'm starting to use as standard

Ghost

This ability lets you escape quickly from your Twilight Shroud and away from enemies, or to quickly close the distance between an enemy to use your ultimate

Exhaust

This ability I've started to use simply because its an excellent way to continue to stay in the fight. I rarely use it with Akali for the slow but rather for the damage lowering effect meaning that when you gank you gank hard

Other Summoner Spells that are probably worth using

Flash

Get out of jail free card. Combined with your Twilight Shroud escaping from enemies is a breeze. Personally I prefer Ghost although this spell is an excellent choice.

Teleport

Due to Akali's fast farming abilities and the fact she can distract people with her Twilight Shroud this is an excellent entry and exit spell.
Use it if you like but remember you can use it on wards to ;)

Cleanse

Since Akali can escape quickly the only thing that can worry you really is being stunned so taking Cleanse is a good way to prevent yourself from being focused. Cleanse + Flash is also a deadly combo.

Ignite

Adds to your burst damage and sometimes can give you that Little bit more damage. I used to use this spell a lot however I find Exhaust more useful now though, use it if you prefer though.

Other Summoner Spells
I don't really see the other spells being of much use for Akali however if they fit your playstyle feel free to use them.

Leveling up your first spell - Your Q - is your first priority. You only need to stick a point in your W spell early on and level it up later. Your E is next priority after your Q but again an early level in it will help with your burst damage

Skill Anaylsis

Mark of the Assassin - Q Spell

This is probably Akali's best ability although it would seem that Crescent Slash does more damage with its higher scaling and damage scaling, however Mark of the Assassin leaves a floating symbol above your opponent's head similar the image used for Crescent Slash This is your mark, and if you physically attack your opponent whilst they are marked they will take the damage dealt a second time and you gain half the energy used to cast Mark of the Assassin back.
Very handy.

Another thing to note is that this spell is ranged, automatically making it superior to her E spell. Since you are a ninja and don't have the burden of mana you can harass with this spell all day long, so do so.

Also remember that when you combine this with the range of Akali's Ultimate you can pick someone off from quite far away

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Twilight Shroud - W Spell

Probably Akali's best move yet the one I see used the worst the most often. So I'm going to go into it quite deeply.

First off the Shroud lasts for 8 seconds and when you are inside it, it increases your armour and magic resistance AND it slows any of your opponents that run through or into it.

Its primary use is obvious. You get in trouble you throw down your Shroud. They can't see you, they can't hurt you. And even AoE moves will do less damage i.e.

However using it well is tough however this handy tip says a lot - http://www.gamereplays.org/leagueoflegends/portals.php?show=page&name=gamereplays-league-of-legends-tip-of-the-week-spectacular-shadow
However this tip was made before Akali was nerfed making it impossible for her to telport whilst shadowed
However when stealthed running in the direction your opponents least expect can lead to a hasty escape

Also since Twlight Shroud slows you can use it to slow down opponents for your time mates to catch or escape from. However if you tower dive with it you cannot be stealthed under a tower however it will increase your armour making you harder to kill

Note that it also prevents you from auto-attacking to reveal yourself so you will have to repeatedly click on someone to attack when cloaked

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Crescent Slash - E Spell

This handy ability scales with both your attack damage and ability power which is why stacking attack damage or going as a hybrid can be effective on Akali.

This ability like all of Akali's has many uses but it has two primary functions, mid-late game it adds to your burst damage when jumping in but it also helps farm creeps.
Since it deals damage in a small AoE it is good versus large groups of enemies and farming.

There isn't much you can saya about Crescent Slash
Apart from one thing, its range is slightly longer than the range of your auto-attack, sounds like a useless hint but trust me, its handy to know ;)

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Shadow Dance - R Spell: Ultimate

Finally on to the juicy stuff. This is what makes Akali Akali. Shadow Dance lets you close the gap between an opponent over an insane range and deal a reasonable amount of damage. Its half the reason Akali is so good at ganking.
It can be targetted on champions or minions so if you are behind enemy lines you can use it as a quick escape tool, as shown in this handy Gamereplays.org tip -http://www.gamereplays.org/leagueoflegends/portals.php?show=page&name=gamereplays-league-of-legends-totw-akali
As long as she can see it she can jump to it.

Now apart from the insane distance, what else makes Shadow Dance so good?
''Akali stores an Essence of Shadow on kills and assists as well as every 25 / 20 / 15 seconds up to 3 total.''
Thats what. Shadow dance deals decent damage and recharges quite quickly so you can use it with your other abilities to quickly kill a target then jump to another straight after.
Note if you use it on creeps make sure to leave 1 Essence remaining so that you can gank or escape should the need arise

The juicy part
Starting Items

First off is a Long Sword and Health Potion. This comes to a neat 450g
So why buy a Long Sword first?
To activate her Passive giving you 10% Spell Vamp off the mark. This basically means that using your Q you can heal up off minions for free, since your spells work off energy. This is much better than the 10% extra damage because being healed is obvious better than extra melee damage

Your next item to buy is your boots and although the item build says to get Boots of Swiftness however you could get any of these shoes Each of these shoes are useful in different situations so buy whichever one is best for your situation. If in doubt go with a Mercury or Swift.

Core early items to aim for

This items are key core items you should aim to get as soon as you have your boots if not before. Choosing which one you aim for first is important to. If you are laning against someone who is unable to kill you quickly then going for a Hextech Revolver for more burst damage and spell vamp then aim for the Rylai's
However I would recommend picking up the Giants Belt if you ever get all the gold at one time


The follow up items
This is where things get tricky. There are a lot of options for Akali and I'll list every option and why it is advised.
First off items that are Great on Akali

Recommended Items
This item gives you more spell vamp, lifesteal and large amounts of AP and AD AND an active ability which you should always remember to use
So stupidly powerful. Getting this earlier is fine because with the passive it gives you almost 200 AP even if it was your First item.
A MUST on Akali
This item gives you AP, passive magic pentration and a large amount of magic resistance. Handy if you are having some major mage problems
Need to take down some tanks?
Since it was reduced in cost recently this item has become a bit more viable and combine it with your Hextech Revolver you easily get over 40% spell vamp very quickly. Which if you do 300 with one hit, is 120HP. Nothing to sniff at
It gives health, damage, ability power, movement speed, several passives and mana is really wasted on Akali
It gives damage and ability power and the more you attack or cast increases you attack speed and ability power. A great item, but you may need to sell it if the game never ends :/

Good optional items
Someone recently pointed out that against a heavy DPS team ulting in then using your Shroud combined with the Active ability of Randuin's Omen you can almost effectively stun the enemy team with the massive AoE slow you will let off
Advised much like the Rylai's Crystal Scepter because it gives you a huge HP boost whilst also adding a slow. However it doesn't add anymore damage to your ability power
It gives health, ability power and magic penetration. Only problem is that if the game progesses you may need to sell it
This item gives a large chunk of ability power as well as magic resistance (also mana which you don't use). With lots of ability power this can hit like a truck. However its a nice to have item as the magic resistance boost is small, its hard to build and you can't use that extra mana. Great apart from that

These items are viable but as it is I don't like them but use them if you wish

Items its not that great to get
It may have great ability power and cooldown but you don't need mana regen or attack speed that much. Better to go for another item
Again nice AP, nice active but you don't need the mana regen. This is an early game item so you are better to aim for something else
A nice item. But you don't any of that mana so you are better picking up a Rylai's because of its slowing passive
Again a very nice item but you won't be aiming to auto-attack that much and the damage can be taken from a Gunblade so you are better off getting one of those

Items to avoid. Full stop
Kage's Lucky Pick Worst of the gold items. And doesn't turn in to anything you really want.
You don't have mana...
Expensive and unhelpful to Akali. Avoid it like the plague
*wince
Under-rated item... but not for Akali.
- All bad ideas

/to be finished

Yet to be added: Item Build reasoning, Overall Use, Good Allies, When to pick Akali