I main Akali and have been maining her pretty soon after she first came out.
I don't have problems with Azirs, you just need to play safely trying to last hit for CS with your Q at range (you can get it pretty easy with practice). Once you get her ult you can do some pretty huge damage and negate his ability to zone you with your gap-closer. Using Twilight shroud to ignore some autos is pretty effective (good for avoiding more damage in exchanges) too since his minions are an extension of his autos, he HAS to have vision of you to AA you or else something is probably bugged.
Trying to pick Akali just to snowball mid-lane is not how you play her either. She's good at 1v1's mid-lane against a vast majority of mages, yes, but she excels when she starts roaming. If you get an opportunity by bullying your opponent out of lane, or by KOing them, then follow up with a gank top or bot lane (whichever enemies are overextending) to net your team some more kills and gold.
If you have issues with too much poke, then trying going for an early hexdrinker which can really help. After that try to rush a Hextech as it is one of Akali's Core items (really I'm quite sure I always get it on her for the most part).
If you are fighting against an AD heavy mid, then an early armguard can be a great option as it provides armor and AP that can give you an early advantage.
Here is one of the MOST IMPORTANT things for any Akali player.
There is a simple combo utilizing her Q. Simply toss your Q onto the enemy lane opponent, then wait for your Q to come up (or just a moment before it does) and then shadow dance to the enemy trigger your mark with crescent slash, toss your second Q on them, and then auto to trigger that one too. This will SPIKE your damage output big-time if you can master this and all the timings. If your opponent is playing foolishly and trying to be too aggressive or not staying a safe distance away from you, then you can easily get this combo off leading to a KO or near-KO.
Learning how to last hit CS is also important for Akali, and knowing when to last hit things with your E or Q is a skill you can learn over time.
Just practice and you'll get he hang of it!