This is Aetherealtenken and I'll be showing you how I play with Akali. Akali is an ap burst assassin and arguably the best chaser in the game. If you're looking for someone that can jump in, kill someone, and run out, this is the perfect champion for you. Also, this is my first guide, SO I may not be used to everything ^^

EDIT: After testing out some more, I find it helpful to grab a revolver before sheen for more sustainability. Both ways work, but revolver trades a bit of damage for more heal.

Pretty self explanatory, the runes give you the ad and ap to start out the game with your passive, enabling you to start off with boots and health pots.

Start off with boots because every champ needs the speed. Being an assassin you definitely need it early. Health pots and your passive should sustain you long enough to save up gold. I like to get a sheen early because it helps you burst out lots of damage. I spent a long time thinking about the right boots though, and find that boots of swiftness is the most effective. Yes, it's viable to go with mercury treads or sorc boots but I think it makes you too slow. To me, rylai's isn't very effective for its cost early-mid game in terms of damage, which is your purpose. If you have Boots of Swiftness and red buff, that should be all you need to secure a kill. Because a rylai's only gives 80 ap for 3150 gold, a rabadon's will give you 140 PLUS 30% ap for 3600. This increase in damage is worth is as fast as possible because Akali doesn't need the hp at this point in the game. The reason why many people rush rylai's early is for the slow and the health. Instead, what they end up with is an Akali that does little damage and with more hp than what is probably needed. After the sheen and revolver, rush the rabadon's for ap. After the rabadon's, get a lich bane which is an upgraded version of the sheen. In addition to the movement speed and the magic resist, you'll notice a significant upgrade in your damage. At this point in the game, people will start noticing how much damage you're doing, and also start qqing about "akali op". This is when you build your rylai's for the slow and hp, and finish with void staff for the magic pen.

Your Q is the primary source of damage, so max it out first. This skill explanation confuses some people at first, but it basically throws a "mark" at your opponent which can be consumed when you auto attack the opponent to do bonus damage. This will be your primary source of damage throughout the entire game, and it's also great for farming minions! Just be sure to proc the mark when you can to refund the energy, or you'll find yourself running out of energy a lot.

The W is Akali's escape skill. It throws down a "bubble" which allows Akali to cloak and gain bonus armor, while also decreasing an enemy movement speed. This is a great ability that lets you stay within range while you wait for the enemy to make a mistake, or you can use it to slow down enemies chasing your team. This ability is really good and the possibilities are endless, but we only need it at level one for now, so just put 1 point into it at level 2. This is also the reason why people by oracle's to combat an Akali. Fortunately, this doesn't really happen early game unless the opponent is stupid, but just burst down the person with oracle's when it does happen UNLESS it is a tank.

The E is basically Akali's farming skill. It scales with your attack power AND your ability power, but this is rarely useful in chasing enemies. If you wish, use it wisely when chasing enemies, because it has a casting time. During the mid-late game, just run into a minion wave and use this to destroy most of their hp, and farm them with ease.

Akali's ultimate is what makes her such a good chaser. This ability accumulates stacks and you can use up to three at a time! You also get a free stack for kills and assists so you can continue your spree~ In addition to doing damage, this ability will bring you to your target, so throw your q, wait a few seconds, and jump to them to proc the mark for tons of damage. Also, this ability can be a free escape if there are enemy minions or neutrals ahead.

Finally, here are some last tips I can probably offer you guys.

With Twilight Shroud (W) you probably want to sit in the middle of it for some time to observe and see which ways your opponents run. Usually they'll pick a direction and you can just escape in the other way. However, if your opponents are smart, they'll stay in the center (unless they're low on health) and wait for you to run out. Remember, the stealth doesn't last very long after you run out, so take your time to think about it. You'll have 8 seconds, and also don't hesitate to throw this down for extra armor or to save a teammate. This skill is probably the trickiest to use and requires the best judgment. If you are unable to run out of the shroud without dying, this is what flash is for! You can flash out or shadow dance if the situation permits.

Another mistake I see Akali players making is using their ultimate the instant after they throw their mark. The mark takes a second or so before you can proc it, so don't jump in right away. Your skill sequence when chasing down an enemy should be:
Q -> R -> Autoattack for mark -> E (ONLY use this if you think you have time, sometimes the enemy will run away immediately and you can just skip this step) -> To finish, use your Q with the autoattacks and use your ultimate as needed. Don't be conservative with your ultimate! you can only save up to three and it's alright to use it quite liberally. You have to find the balance, however, so don't use it rashly either. And also a note: Akali really starts to prove her worth if you have a good team, if your team doesn't really know how to play with an Akali, things can turn out pretty bad.

If you have any questions or if i missed something, then feel free to comment below and tell me. Sorry if this was messy, as this is my first time writing a guide.
^^ Thanks for reading.