Ok i know it looks a bit weird at first, but bare with me. I found the best strategy with Evelynn is to stay back a little bit and poke with your Q ability.
Always wait for her passive to become active, but remember champions can see you. so try to Stay safe until you can rush in and kill.
when fighting a single player try to start with W, and open with E.
if you have your ulti (and necessary) use that, otherwise spam Q like no other and get in E when you can. If they run, W right away and try to chase them. (if your Q isn't connecting stop chasing)
i focus her on a balance of Ability power and attack speed/damage. when your E is used she can generally kill pretty fast.

during team fights i always focus on the squishy targets, and tend to stay away from combat until i have a good opportunity to jump in and take someone down.
Ironically i noticed she gets focused a lot! I'm not sure if it's cause she's squishy, or because she's deadly, so try not to stir up to much focus on yourself and stay stealth.
A lot of people go for Rabadon's sooner, but i generally try to do more in damage and speed vs abilities. I KNOW her Q is her 'cake taker' but he ability power isn't so much further ahead of her attack damage that you should focus on it.

i know this isn't for everyone, but i think it's a fun mix to play her as. She is QUITE squishy in this build, but when you're running in and killing, and getting out quickly, it's an ok drawback.
(Remember, this is how I!! play her, it might not be for everyone, and might not be for you. so please don't hate on it if you prefer her another way. I do very well with this build)