1) Your Q will remain on an enemy target longer than its cooldown, so when trading, land a Q, wait for it to come off cd, go in and proc your first one, then immediately Q and proc the second.

2) Your Twilight Shroud provides vision, so if an enemy flashes over a wall, throw your W over and ult to them. Alternatively, if you are being dived in mid, W over the wall to Raptors and ult to safety.

3) When tower diving, wait for a minion wave first, because then once you have killed your target, you can simply ult back out to the minions after.

4) Normally hextech is your only item with AD, but don't be afraid to invest more heavily into hybrid. Vs a veigar? Take QSS, and upgrade into merc scimitar later. Don't want to continually spend money on an elixir? Take red pot which resets by 30s after each kill (if you're fed these can last almost endlessly, so are super gold efficient!).

5) Void staff does more damage than deathcap on Akali. I don't just mean it's more gold efficient, I mean it actually provides you with more damage to champions, so get it 100% of games!

Bonus itemization tip: People always argue over deathcap vs lichbane (assuming you have hextech, boots, void, zhonyas and a def item). Deathcap will give you the most damage, so take this most of the time, unless you are playing top akali with TP, where you might want to go lichbane for split-push presence. Alternatively, get no def item and take both.

I am not sure how much of this is basic knowledge and if this is actually useful, just wanted to share something I learned by playing a lot of Akali.

1. Mark enemy, wait until cooldown comes up, then go in. This works because the mark lasts longer than the cooldown therefore you can activate 2 marks. I recommend activating the first one with autoattack and the second one with E, since you are going to be on the move after your combo and don't want to stick for the long autoattack.
2. Use your shroud to chase or evade enemies. Since the latest changes it gives movement speed to Akali so whenever you are ganked, just pop the shroud for extra speed. You can also drop it in the path of enemies chasing you since it slows them a bit. Be careful thought because Shroud needs a lot of energy.
3. If you are on low health in lane, use Q on minions and autoattack it after. Akali has built in spellvamp and generally builds gunblade first so you will have decent sustain. Be careful because some more experienced players will go for a trade after your Q cooldown is up.
4. Save your ult charges when needed. In lane it is okay to always have 2 charges, but don't spam it too often. All your main damage comes from Q mark popping and focus on that. If you can pop it with autoattack or E, do it. No need to waste charges if your enemy is close because he might flash / dash away the next second and you will have to wait the cooldown.
5. Be patient. This is very important as playing Akali. Don't go all in on the tank, just wait for either of the team's initiation and wait for the important CC abilities to go down on your teammates. After that find the best target you can burst down, then go for the next one. This is where Zhonyas comes into play. In Zhonyas you can wait for all your abilities to come back and for your team to back you up. Sometimes if the enemy tank has extended too far, you have to focus him down with your team but other than that always go for low hp targets.

I hope I helped someone to master the champion since she is one of my favourites. Please forgive my basic English. Give feedback and add your own tips or correct if I got anything wrong. Thank you!