Akali is an assassin with strong traits of a fighter. She is an hypercarry that relies on having a strong early game, and keep that advantage to seal the game before the very late game arrives. For an assassin, she has not the highest burst: it cannot compete with 'pure' assassins like Talon and LeBlanc; but it's compensated by one of the best dueling potentials of the game: in a 1v1 fight, if she survives the initial burst, she will pretty much kill any champion.

Akali is an "easy to play, hard to master" champion: despite her reputation as a 'noobchamp', when playing against skilled players a mediocre Akali player will fail badly, and only high-skilled players will be able to keep up.
Akali is one of the best champions when it comes to punish mistakes (especially when it comes at positioning), and that's why she has such a bad reputation in low elo: when players are constantly making mistakes, Akali will find very easy to abuse that situation and get fed.

As Akali, you should be able to calculate damage properly, as there are a lot of factors involved: your spell vamp, which is as valuable as your damage late game, and the cooldowns of your opponent since you will probably need, at least, two rotations of spells to kill your opponent. You should never engage in a teamfight, instead you should wait for both teams to engage and then enter quickly to take down priority targets (usually the enemy marksman or mid) before they can react. Once a priority target is down, you should be able to survive the teamfight simply because of your high sustain. Keep in mind that you are really squishy actually, and your sustain is the only reason you don't die instantly when you enter a fight, so try your best to dodge crowd control effects.

Akali's main position is mid, although her dueling capabilities allow her to play top without trouble. She's probably the weakest champion pre-6, you'll have to learn how to survive that horrid phase. Once she reach level 6, it's when she unleashes her whole potential and thus when she can easily take the aggressive side against most champions.
From 6 to min ~20 is when the most important phase of the game for you takes place: you need to abuse every mistake (like being squishy or not having CC, which is a mistake by itself against Akali) and get ahead. Once you are ahead, you can roam the map hungering for kills, helping your teammates to take objectives and trying to seal the game quick. It's really important to get ahead, since she's really fast or famine: if the opposing team locks you down early game, it's very probably that you will be behind the whole ge amand, without nothing to offer but damage, you will be basically useless.

Jungle Akali is viable, although it's important to establish that there is no reason to pick her over any real jungler other than the fact that you enjoy Akali, and I hope you do (that's not an option). Her first clear is really weak, you'll have to do it very carefully and you will be really low on health. Once she builds her jungler item, though, her problems dissapear and she can take camps quickly and safely.

+ Strong, reliable burst.
+ Has one of the highest dueling potentials.
+ Easy to get the basics, but a high skill ceiling that ensures you'll learn a bit more every game.
+ Strong against the vast majority of popular mid laners.
+ No mana.
+ Likely to make your opponent ragequit.

- Probably the weakest champion pre-6 in the whole game.
- Very feast-or-famine
- Easy to get the basics, but a high skill ceiling that ensures you'll need a lot of time to master her in high-elo games.
- Likely to be called a 'noobchamp'.

Marks

Greater Mark of Hybrid Penetration: This is the preferable mark. Your Q + aa combo it's the most important early game, and your E deals physical damage, that's why the armor penetration is so valuable.

Greater Mark of Magic Penetration: This one is OK if you can't afford or simply don't want to buy Hybrid Penetration Marks.

Seals

Greater Seal of Armor: Your go-to seals. As the weak melee champion you are early game, every single ranged champion (even Anivia) will sneak in basic attacks whenever they can, and those 9 armor can help you prevent more damage than you think. If your opponent happen to be a melee champion, he will probably be a heavy AD caster such as Zed or Jarvan IV, where Armor runes are mandatory.

Greater Seal of Scaling Health: If you are not against an AD caster, these are a pretty good alternative to Armor ones. In fact, they are more popular for most mid laners right now.

Glyphs

Greater Glyph of Magic Resist: The standard ones.

Greater Glyph of Scaling Magic Resist: If you are not facing an AP champion, you might prefer these.

Greater Glyph of Ability Power: You give up survability for a stronger early game. If you were not the weakest pre-6 champion that'd be an interesting idea.

Greater Glyph of Scaling Ability Power: Like the previous ones, I personally dislike them. You give up survability for a stronger powerspike mid-game, which in most cases you don't need.

Quintaessences

Greater Quintessence of Ability Power: Pretty standard runes for damage. Personally I don't see a reason to play any other quintaessence.

Greater Quintessence of Scaling Ability Power: They become stronger than the flat ones at level 11. I prefer the little help you get early game from the flat runes, but these ones aren't bad at all. It comes down to personal preferance.

Greater Quintessence of Movement Speed: Never tried this one personally but I've seen people with them. I can't tell too much.

Cunning

Tier 1: Wanderer over Savagery because it allows you to roam more effectively. If you need some help to last hit, or you don't want/can't roam, you can pick Savagery for the extra damage to minions.

Tier 2: Runic Affinity is useless because we won't be taking any buffs, and even if we were, we don't need that extra durability. Secret Stash just can't compete with Assassin : it gives us some increased damage in lane and when hunting targets. As an assassin, we will never decline having more damage.

Tier 3: I think I don't need to explain why we don't need Meditation . Also, Merciless is just too good to give us that extra bit of damage when we are short of burst.

Tier 4: As before, Bandit has no effect for us, so we'll pick Dangerous Game .

Tier 5: Intelligence is not a bad mastery, BUT it has the con of being paired with Precision: That mastery is probably the best non-keystone mastery of the game, as it alone offers more penetration than Battering Blows (1-5-1) and Piercing Thoughts (1-5-2) together to any target that has not stacked a lot of resistances. It's also one of the reasons why people pick Thunderlord's Decree so often.

Keystone: Thunderlord's Decree is mandatory. The amount of burst it gives is insane at all stages of the game. It's important to note that a basic attack proc'ing Mark of the Assassin counts as an attack and a spell, thus hitting an enemy with Q followed by an aa triggers Thunderlord's Decree . Against melee champions, this helps a lot to burn your opponent's health slowly and win really short trades.
Stormraider's Surge has no point for us since we already have Shadow Dance, which we will be using constantly at all stages of the game.
Windspeaker's Blessing is useless for us, I'm sure you figured out why.

Ferocity

I prefer this tree much more than Resolve for Akali, since you are supposed to be aggressive and I think you only want defensive masteries before level 6. Just in case you want to pick Resolve, I'd recommend you to go for Recovery , Tough Skin , Runic Armor and Insight .

Tier 1: Sorcery over Fury , for obvious reasons.

Tier 2: Double Edged Sword is just too good. You increase your damage by 3%, while only taking 1.5% more, which you compensate by the extra bit of healing you'll get from dealing more damage. Feast is not bad; indeed it's pretty good, it just has the problem that it competes with Double Edged Sword

Tier 3: This one is up to you. I like to grab Vampirism for the bit of sustain it gives you early game, as opposed with Natural Talent which is irrelevant early game but [usually] better late game. As you are supposed to be ahead and snowball, I don't think you should need a bit extra damage, while those 2% life steal/spell vamp can help you in tough trades.

Tier 4: Oppressor because priority targets are expected to be the target of most CC in your team, so you should be able to take advantage of this mastery with relative ease. Bounty Hunter is too situational, and you won't probably kill 3 different champions consistently early game to make it worth it.

[Work in progress...]

[I'll update this soon I promise as long as I have some free time D:]