Great question! Evelynn is lots of fun. She is the definition of assassin, but that also means she has a fair amount of skill required. Her stealth is of course essential for catching targets out, but it is also very important to remember that being stealthed also grants her bonus mana regen. Having cut your teeth with her on URF I bet you weren't use to how quickly spamming her abilities is able to drain a good portion of her mana. She actually isn't the worst at this, but it is still important to back off and remain in stealth for a while to build your mana back up instead of going ham & cheese all the time. Still, it can be useful to build at least one mana item, especially if you are planning on building CDR.
Any of the
items are always good options.
By far the most common option for Evelynn is AP. She has good scaling with AP on her Q and it is the easier ability to land when kiting or chasing enemies. The fact that it deals AOE damage makes it a good option for farming and jungle clears. Also applies a bit of bonus AD scaling, so keep that in mind.
is good on AP builds because it gives her good sustain and her Q applies spell effects, so it helps her with her chasing and kiting.
I really like her W because it gives her some of the best bursts of speed. Paired with boots and items that give bonus movement speed, she can be all over your enemies and it is very difficult for them to escape.
&
items typically work well because of the move speed, depending on which stat you are looking to max.
And don't forget your
, take Alacrity for AP and Furor for AD builds.
Her E does also scale a bit with AP, but is really a better focus for AD builds because of the attack speed steroid. I really like focusing on-hit effects with AD Evelynn because you can quickly burst enemies down. This is less common because it is really more of a challenge. You absolutely must be ON TOP of your enemy and stick to them, but you can absolutely shred their HP.
is great because it gives her an AOE on-hit cleave and an auto attack reset.
As for her Ultimate, .. I honestly don't rely on her ultimate as much as I do with other champions. It does %HP damage, so always good, and obviously scales with AP, which is most likely why AP is more common, but the slow and shield are definitely useful. Basically, use it whenever you can in order to further 'lock down' an enemy who isn't prepared to get their face scratched off. It can be useful when you are really wanting to dive into your enemy team and pick off their ADC. Try to land the AOE on as many enemies as possible because your shield's duration lasts for 6 seconds for every enemy hit, and you're going to need it in order to get in and get out before you get killed.
Remember, your stealth ONLY works when you are not revealed by true vision, (TOWERS) and when you are not too close to enemies. Look out for the vision warnings that will appear above your head.