All of you players looking to make the most of Akali... welcome! I'm Esaelon, a currently unranked player but one who sometimes overthinks things and has a strong math background. Based on my dissatisfaction with many builds which too often resulted in Akali getting marginalized, I've crafted my own custom build, which you see here. I'll explain why the build is how it is, and a few ideas on how best to use her. This is my first attempt at a guide, so please rate and comment, and most of all... Enjoy!

Pros:
Massive burst damage and snowball potential!
Bonus spell vamp due to her special
Good chase using shroud (slow) and shadow dance (dash)
Shroud gives mild cc (slow), and defense and stealth to Akali (armor/MR to Akali while in it, stealth except for 0.5s after each attack)

Cons:
Melee, too easily countered on mid lane; often needs gank support until 6
Some characters can outburst her (LeBlanc), outrun (Cassiopeia, Yi), or outharass (many)
Can be countered hard by characters that can reveal (Lee Sin), knockback (Lee, Xin) or hard cc
Stealth countered by snares
Squishy (spell vamp helps, and shroud helps tremendously, though)
Countered by revealing items (Oracle's, Vision Wards)
Once burst is gone, she can be virtually harmless (being manaless helps, but is not enough: blue buff can be very helpful both for keeping enough energy and reducing her low cooldowns allowing her to burst harder)
Expensive: requires runes to be really effective
Can be marginalized (though this build makes it far more difficult)

Thanks to the recent buff to Akali, she is no longer requires a perfectly customized rune page. Rejoice! Her two innate abilities, Discipline of Might and Discipline of Force, which increase based on bonus AD and AP, no longer *require* bonus AD and AP to activate.

Because of that, I have revised the preferred masteries and runes for her.

However, all of Akali's damaging abilities (Q, E, R) scale with AP. The scale ratios are not very high compared to their flat damage, which means that Magic Penetration gives her a lot more bang-for-the-buck early on. Because of this, my current rune page is justified by the following reasoning:

Defense: Seals, 5 scaling HP. While defensive runes (flat armor or MR, scaling armor or MR) may provide more effective HP, this is cut down by APen/MPen, while HP is not. Given any opponent worth their salt will build APen/MPen to boost their damage, the runes won't give enough to matter without building items for defense also, and Akali needs damage. Unless enemy is using oracle's or wards, most of the time you need to avoid getting hit anyway. Akali can never be truly tanky, not and get kills. Fortunately, Akali is melee, so she does innately have scaling MR.

Damage: Everything else. Yes, seriously.
3 scaling AP glyphs.
9 MPen marks, 6 MPen glyphs, 3 MPen quintessences.

Because of the way magic penetration works, and Akali's high flat damage and relatively low AP scaling, this gives her far more damage in early game than maximizing AP. I'll put up numbers later. Sorcerer's shoes alone will allow her to do true damage against those ranged opponents foolish enough to not build items or runes/masteries with MR (all champions have, as a minimum, 30 MR). Melee champions have scaling MR as well - except Kayle and Nidalee, who recently lost their scaling MR to a nerf.

Changing this was difficult for me, but realizing that Akali is mixed-damage, I checked how her damage compared with giving her the APen% mastery... and it helps, more than 6 AP of Mental Force or the 2% damage of Havoc. To minimize early game squishiness, this build is now 21/9/0, with more MR on defense than armor (for mid lane; for top, it may be better to reverse this). A possible alternative would be to give up Archmage for Sunder and forego Weapon Expertise, perhaps for Havoc, though it is questionable if this would be good for Akali's mid-late game.

This build is primarily for mid lane. Unfortunately, anyone who has played mid Akali should know there are many mid characters that can counter her, and some counter her hard. Nearly any champion with range can harass Akali to death (though Akali is stronger than some, thank goodness), so Akali needs to be careful in selecting her abilities in order to maximize her ability to farm early on, but roam and gank/kill when she hits 6.

In order to counter ranged champs, harass, and pulls, Twilight Shroud is often the best first pick. You may not be able to keep up on CS, but you will often be able to avoid starving completely. Do not be predictable in your movements when in the shroud, either. This works better or worse against some champions: LeBlanc, Talon, Orianna, and Teemo are particularly difficult for Akali, and Lux is not easy either.

If you know you will not be facing those dangers, it may be worthwhile to postpone shroud, perhaps until as late as level 4.

Mark of the Assassin, her Q, is her primary method of harass, helps with farming, and keeps Akali's energy high if you proc the debuff, which can be done with her autoattack or her Crescent Slash (E). If Shroud is unnecessary, get this to harass your opponent and deny their farm.

Crescent Slash is typically taken at level 4 if shroud taken early, and at level 2 if shroud taken late. The main problem with it is it is easy to go low on energy when using it frequently, and even more if using shroud. Furthermore, it does physical damage, so it will not benefit from magic penetration. It can be useful to take it earlier to help in farming, though it tends to make her harass, and ability to more easily last-hit, weaker. Level 3 can be better if you are having serious difficulty farming even with Shroud and the early Mark would help more.

No matter what, get level 3 Mark by level 5, and max it by level 9. The second skill to max depends upon your desired style: Crescent Slash helps with farming and burst damage, but not tremendously on the latter, while leveling up Twilight Shroud gives additional armor and MR while you are in it, which can help some, though not tremendously, in surviving battles.

Her best combo for kills after level 6 is Q+R+E or sometimes Q+R+AA, or Q+E when you can stay close and do not need to use R or cannot at the time (out of charges).
The Q+R+E (repeat) does massive damage: even at lv6, it does 220 + 160% AP + 60% AD, most of which is magic damage. Akali can do it as quickly as she has essences of shadow and energy available (3 if not used for awhile, even more if she can get kills/assists along the way).

You will be a squishy assassin, but there is not much escaping that for Akali... expect to get focused! You are the cleanup crew, however because you unstealth when attacking, leveling up shroud after level 9 gives you a little more essential durability.

I usually go with Ghost and Exhaust. Ghost is helpful for getting here into, and out of, battles fast, and can also help with split pushing later on. I have never managed to make Flash work on her, but in theory it could. I just find Ghost more flexible and useful for her.

Exhaust, with the Mastery, is amazing. Reduces movement speed, attack speed, damage, armor, AND magic resist. Perfect setup for a burst-down.

Ignite can work better against heroes with massive regen/lifesteal/spellvamp, such as AP Master Yi, Dr. Mundo, Sion, Nasus, and Lee Sin.

While you can take Exhaust+Ignite for almost guaranteed kills, realize that you will have little escape pathway (though, creative use of wards and Shadow Dance may on occasion work).

Thanks to Akali's high flat damage but relatively low AP scale, magic penetration is very helpful, so it is good to have at least 30 flat MPen as soon as possible: with runes, you can get near or above 15, and the next 15 comes from rushing to Sorcerer's Shoes.

Next is boosting Akali's already high burst, and Lich Bane is the solution. To be stronger against champions with less armor, get Sheen first; to be stronger against those with less MR, get Amplifying Tome and Blasting Wand before finishing Sheen. Once Lich Bane is finished, be mindful of it when attempting to secure kills. It can be worthwhile to wait briefly after Shadow Dance for Lich Bane to come off cooldown before using another Q or R. You should be able to land at least one Lich Bane proc per Shadow Dance charge.

Spell Vamp is next, enhancing Akali's innate ability. The Hextech Gunblade is the best source for Akali, as the active makes it more difficult for enemies to escape, while the 45 bonus AD it offers translates into an additional 7.5% Spell Vamp, for a total of +27.5%. Combined with the 12% Life Steal, allowing her to heal off creeps, and her sustain goes up dramatically. Furthermore, this lets her push quite well (and even better with a Void Staff).

The next item depends. If the enemy team is smart, then they will build MR to reduce your TONS OF DAMAGE, and then Void Staff is your next choice.

Alternatively, if they do not build MR and you start getting lots of kills, a Soulstealer can accelerate your snowballing, but only buy one if you can feed it! This is almost certainly even less viable thanks to the recent nerf of her Twilight Shroud, so it may now be completely limited to pubstomping, so the next item is a more typical, safer choice.

Eventually, you will need to get more AP, so Rabadon's Deathcap should be purchased, as it will amplify whatever AP you have, and is gold-efficient when you have at least 7 AP from other sources.

If you did not get a Void Staff before the Deathcap, get one immediately after.

For the final item, if the Soulstealer is not viable or has fed you sufficiently that you have the gold to replace it, there are three main alternatives:

1) More Spell Vamp: Hextech Revolver, to Will of the Ancients;
2) More burst for the first kill: Needlessly Large Rod, to Deathfire Grasp;
3) Lastly, a better chance for escape: Needlessly Large Rod, to Zhonya's Hourglass.

There are no other seriously viable alternatives: with Akali's massive damage, these options give better survivability than any items that offer more tankiness. One popular item, Rylai's Crystal Scepter, has potential roles, however for its price it usually offers too little damage for Akali. One of the Enchantments, Alacrity or Furor, are much less expensive and can often serve the role better, without eating the item slot. Rylai's often turns Akali into a supporting role, rather than a clean-up crew or, what she snowballs into, an assassin that can win 1v2 with near-full life and can do well in 1v3, and push quite hard to boot.