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Having played Evelynn for a long time and tried all sorts of builds I always come back to these items when playing AP:
Sheen->Mobility->Mejai->Lich Bane->Giants Belt->Rylai's->Deathcap->Guardian Angel/Banshee.
A early Sheen works wonders because the amount of HP or lack thereof on your enemies upon acquisition means that a +100% base damage for a single attack is very notable. Use that as early as possible, try to do coordinated ganks because the difference between two successful or failed ganks is really notable and can even make or break the game for you.
Squeezing a Mejai in before Lich Bane is because Lich Bane is not that much more effective than Sheen is before you get close to 200ish AP and it is expensive. Mejai is far more cost effective and you want to fuel this baby early on. Many dislike snowballing with Mejai but granted the huge potential on Evelynn I say you should never leave this baby out. After you get Lich Bane you need to be very careful because you are very vulnerable but at the same time you hit like a truck. But since teams are grouping up at this time you need survivability to last long enough for your ultimate to kick in. Giants belt works wonders and Rylai's more so. The snare effect of Rylai's is not a must but it is nice to have, it's more the HP/AP combination you want.
The game is usually over by now but if it drags out you want that GA for a nifty passive and generally delicious defensive stats. Banshee has a interesting feature that once you pop out of nowhere with a bubble on you they can not reflex stun you giving you a little extra time to do your thing.
So be aggressive and eager but also be smart. Someone asked when is the best time to gank and, yes, it would typically be at level 4ish and best when you have your Sheen.
Go for it! A 500-600 AP Evelynn will tear people up but failing early will gimp you the entire game so play smart!