Akali is a difficult Champion to play as optimally. I will do my best to make it as easy to understand and short as possible. Also I didn't put anything on synergies, as Akali doesn't really synergize with anyone, as she just kills people fast. If you have someone with sheild, heal, cc, or damage boost that great, but don't beg your team to be someone in particular becuase then they will hate you.

Passive: When you hit a champion with an ability, a circle appears on the ground,you gain movespeed towards the outside of the circle. When you leave it, your next auto attack gains increased range and damage. (Lots of Akali's damage comes from using this passive as much as possible)
Q: You shoot out kunai in front of you. This is your range as Akali. You use this to poke and farm. (If you hit someone with the tip of the hitbox, it will also slow. You can see this hitbox when holding the ability down.)
W: Put down a smokescreen. You gain a big movespeed bonus, refresh your energy, and also are invisible while in it.
E: Flip backwards and throw a shuriken forwards. You can recast to dash to the target it hit doin more damage and allowing to kill them.(You can use this over walls, and it will proc your passive very easily. It will also leave a marker if you put down a smoke screen and throw it into it. You can dash to the marker by using E again.)
R: Flip behind your oppontent and deal damage. After a short time, You can use it again to dash in any direction and deal massive execute damage. (You can use the dash as a finisher or an escape if needed.)

Akali is an assassin, you need to hit and run. Use her combos in ways that allow as many passive procs as possible. Poke your oppents with Q while farming, and all in with R. It is not hard to understand and do, but difficult to master the timing and passive procs. You want to snowball as much as possible. Kill your opponents beofore they can even notice that you are there.
CC is your worst nightmare, dodge it all costs and if the enemies have loads of it, it may be worth bringing cleanse or even QSS.

You suck early, and will probably lose lane to a ranged mid laner who understands that range is their friend. You want to try to poke them with Q if they walk up to get a minion, and if they are low HP, kill them. I will go over that part later. If your Jungler is fighting someone nearby it is definity worth going in on them to stack Ultimate Hunter and Eyeball Collection(or dark seal if you have it.) Early game as Akali is pretty boring, unless their laner decides to walk up to far, then you can just dive them and easily beat them as long as they don't CC you. If given the chance to all in them, and your HP is fine, just ignite them and be sure to slow them with the Q sweet spot.

Now you should have Hextech Rocketbelt, and be able to roam pretty easily. If you can manage to kill you lane opponent, just roam to try and get some more snowball potential. Probably go top first to avoid the CC the bot lane support probably has.

By this point you should have many items and hopefully snowballed. If you see someone alone, it may be worth killing them by jumping out at them. You don't need to spend more time walking around to surprise them. You can just walk up to them if they have no escape tools. You can catch up by useing E to hit them, or even using it to flip towards them. You can use Q to slow or just ult with Hextech Rocketbelt to get it for free. You typically would rather 1v1, but in teamfights, you can just kill their backline easily.

This is the most important part of the guide, becuase without combos, you can't kill people.
Main Combo: This is your bread and butter.
Use R to dash onto your target, and use E the second you pass them so that you cannot miss it. The E will proc your passive when you flip out of the circle, so dash back in and auto them. Use your Q to deal extra damage to them, or even slow them if they run. While you are using your Q to deal extra damage use your W to gain more energy and movespeed to make sure you can proc your passive as much as possible. The moment you can use your R again, use it assuming they are low enough to be killed by it. Congrats, you just murdered your first person!
There are ways to change the combo to make it do more damage over time, but seeing as this is meant to be a very basic guide to get you through your first Akali matches, I will not go over them her. Just note that if you can use passive more, then do it. It does so much damage during fights. Also try to auto attack as much as possible to use Lichbane. (You may also build Nashor's Tooth if you are struggling to auto in between abilites. Nahor's will also help increase damage from your passive.)

Note that unsing Rocketbelt or Flash to easily dodge is a good idea assuming your E is on cooldown. Try to use Rocketbelt to get in and Flash to dodge if possible becuase the projectiles from Rocketbelt to pretty good damage.