Welcome to my first guide!
Quick introduction. I have picked Akali about level 12, when I encountered a player, whose nick I forgot to write down. Me and my 3v3 team were practising on custom mode (we all were like just above level 10). We hosted a game, and what we got, was an enemy team which consisted all of 30 lvls. Akali and Mordekaiser owned us(sadly I don't remember who was the last champ, but seems that it was Sion). I decided to give her a try. It was hard to collect all the IPs (I wasn't really good at this game at that time). After buying Akali I had to understand what's going on with her abilities, and after doing so I realised, that Akali can give a lot of fun. I liked her so much, that it was the only champ I played until I reached 30.

Recently I decided to write a guide, and share my view of this champ. Share my tactics. I do not claim, that what I write here is the best way of using her, but I think that's the enjoyable and rewarding way.

I think that pros and cons of this char are written here in so much guides, that there's no need in writing the same thing another time.

Nothing new or revolutional in this chapter.

Runes.
Rune setup is quite obvious and allows you to get a slight advantage early game. I prefer HP seals over dodge because I do not like to rely on luck, but if you wish, you can swap those. Magic penetration allows you to hit harder and ability power over 20 at the start of the game let's you activate your precious passives at level 1(what was been written in numerous guides).

Masteries.
Again, a basic setup, which allows you to get doran's blade at a start of the match instead of longsword to activate your passive.

Summoner Spells.
I choose exhaust and flash. Why? Because they both have many uses, and exhaust let's you to catch up this annoying enemy or get rid of a chaser when you want to get out. Flash has many uses. From escaping from unwanted attack to extending range of your ultimate skill.

Akali's skills is what make her awesome.

First up her passive:

Twin disciplines:
She gets spell vamp when reaches 10 bonus attack damage, and her attacks deal an additional magic damage when she hits 20 bonus ability power.
It is important to have activated BOTH passives early game, but I tend to focus more on the second one. Why? Reason is simple. Let's assume, you go AD with her. Is that 50% of spell vamp really going to save you when you have like 100 ability power? No.
Activate first passive, but let it stay at 10%+ hextech revolver bonus. Then focus on ability power.
More ability power = more skill damage = more heal from spell vamp

Mark of the Assassin:
Your basic skill. Skill that gives you kills some might say. It's good. It hurts to get hit by it. But I'd not risk to say, that it's the skill that gives you kills. But max it ASAP.

Twilight Shroud:
That's the skill that gets all my love. The most awesome part of Akali in my opinion. It's great as defensive (hiding etc) or in offensive (great psychological weapon!). It has many uses. It is certainly most versatile skill Akali has access to.

Crescent Slash:
Fine skill. I generally use it if my main combo is not enough or to get rid of minions after the fight. In late game you can use it to trigger lich bane buff and for example solo a tower (be careful with it though).

Shadow Dance:
It's what makes Akali so good chaser. Remember to have at least 2 stacks of it when you're about to start a fight because of the (quite big) possibility that enemy may survive your initial combo. When fighting in a proximity of minions you can target enemy minion to get away fast.

A lot of guides suggest that standard combo should be: Q-R-autoattack-Q- autoattack -E etc.
but I find a way more useful and safe this one:
R-Q-autoattack-E-Q-autoattack (in case of getting in trouble or enemy escaping put R-skill in the appropriate places)

It's standard item build that focuses on AP (remember that you shouldn't sell doran's blade until you have hextech gunblade). What I'd like to focus on are Mejai's Soulstealer and Abyssal Scepter.

There's no room for mejai's in my build. You may ask why. Answer is simple. You can have enough damage without it, that's the first thing. The second, and more important is, that people, who collect stacks get targeted a lot, and that's the thing we are trying to avoid. Of course, you can buy it, I even often do it, when situation needs it. It's easy to get stacks, but really, you don't need them to succesfully dominate the battlefield.

Abyssal Scepter. Reduces magic resist of nearby champions. Cool. Really. You can contribute to team in a different way that scoring kills. And it gives a little bit of magic resistance too, which can be needed in some cases.

That's the most important part of playing as Akali. Tactic and thinking. You need to carefully choose your target. Do not initiate carelessly, and always have in mind at least two possible routes of possible retreat. It may seem hard to do when you are in the fray, but it'll come with practice (oh right. Play a lot of practice games. A LOT). I'll show a few tricks I like to use, but there are endless possibilities.

Remember to always have twilight shroud nearby. And cast it often! Always when you fight, you want to have it near you (do not sit in it all the time- when you do, you are easy to target. Make advantage of your disatvantages, be there, where enemy does not expect you to be and strike the most vulnerable opponents. Sounds easy. And indeed it is easy, but you must practice. You must learn to have a wider view of the map, than what's currently on your screen. Do not chase too long, sometimes it's better to have enemies running away, than dead.
Remember. You choose when to fight, and who to strike, and don't be afraid to ks. It's teamplay and everyone should understand that it is your role and you try to do your job. You are not as good as other characters in other tasks.

Get a few deaths. Really. What you do not want is to be targeted in every gank possible. Try to have as many deaths as your team. It'll make your job easier.

Offensive use of flash. You can extend range of your Shadow Dance by using flash just before casting ulti. Sometimes it can mean death to the opponent (combined with the slow provided by rylai's crystal scepter).

Casting exhaust. When you are outnumbered it is sometimes needed to cast exhaust not on your target, but on that pesky friend of him, who ruines your attack by helping his buddy. In most cases it's the way I use this spell, because... I do not find myself in situation which can force other uses (of course when you participate in gank it's crucial to use it- make sure, it's ready to use!).

Of course it is the ideal scenario, which of course never happens.

Be aware, be quick and deadly. That is the most important part. And of course have fun!
Thanks for reading. If you have any questions add me on LoL:
Scarlet One