When you start playing take a look at the enemies roster and decide what you will be needing most Armour, Magic Resist or Magic/Armour Penetration. If you feel like the opposing forces are inadequate to deal with your burst damage then focus on getting more Ability Power to ensure quick and easy kills.

When starting in lane play passive until Lv 6 because a majority of your kills will be given from initiating with the ultimate as opposed to running at the enemy with the other abilities.
If you must engage before Lv 6 try using a Q to E combo to stack the damages this will wipe alot of the enemies health early game while it wont be enough try backing off until cooldown is up then follow through with another Q to E combo. This method will often result in an easy kill as long as you are patient with it.

Akali's abilities have great synergy with each other so ensure to use combos as much as possible rather then focusing using only a single ability.

Attacking 1-2 people - When going in for the kill focus the champion with the most damage output. When you have your focus, toss a Q to initiate then follow up with an R to E combo to gap close then stack the additional damage from the tossed Q and used E attack this ensures damage is dealt and fast. If the Champion is either running away or still alive putting up a fight don't be afraid to use another R followed by a Q to E combo when the cooldown is up.

Team Fights - While Akali does big damage always let a tankier champion initiate then use Akali to follow up to output her damage. This way you are not focused with a majority of the abilities keeping you alive longer to continue the big damage output.

Evading - If you are in a battle that is going sour throw down your W and try splitting off into a diagonal direction back to tower or into the jungle if its safe to do so. I prefer to duck into the jungle if I find it safe to do so that way its an easier split because enemies often expect a dash to tower while in jungle you can jump to monsters over walls to separate even better from bad engagements.

While Akali does some hefty damage it is important to know that you must play passive and patiently until you hit the pinnacle of damage output which I find to be right when you hit level 6 or 7 and the first major item (Hextech Gunblade) is built. If you get shut down early get help from your teammates to get even on the score board as your opponent because if Akali gets left behind she will be useless until caught up again. Granted if you end up a kill or a few hundred gold behind it wont be the end of the world however if it starts to get a little ugly like 3 or 4 kills behind you will need some major assistance to get back up there.