Hi. New to Mobafire, and this is my first guide.

Written because I love the new eve, and want people to respect her again.

I've had the most success with Teleport and either Exhaust or Flash.
There are always other options.
Surge is good, but falls off late game.
Ignite, also good, but not a great fit with AP eve's play style
Anything else, well... if you make it work, good for you, but I'd suggest sticking to the toolkit.

Evelyn's Q (Hate Spike) is on a 1.5 second cooldown, at a .45 AP ratio. Throw a little cooldown into the mix, and this is a consistent, high output killing tool.
Take a point at level 1, then start maxing it from level 4.

Hate Spike is now a linear ability, and can hit multiple targets, so remember to POSITION yourself to take advantage of the direction it fires in.

W (Dark Frenzy) took me a little while to figure out. Sure, its a great move buff, but its just a little underpowered. Then, amidst a mushroom and cleaver storm, I realised the true utility - jumping out of slows.
If your opponent has a slow, HOLD OFF on this ability as long as possible, and you can completely counter their slow.
I'd suggest taking a point at level 2 and maxing this second if manoeuvrability can help you win the lane, or at level 3 and last if its not so helpful.

E (Ravage) has a very short range, and while the Attack speed buff is nice, its just not the point of this eve. It does have a nice single target damage output, though, so take a point at level 2 or 3, and max third (unless its more useful than W)

Finally, the ultimate - R (Agony's Embrace). Again, it looks fair, 25% magic damage based on max heath (nice), but underwhelming. Magic resist, or flashing out.
Wait, flashing out? NO!!! there is NO DELAY, its an instant effect. There is almost no way that you can fail to land it. Snap - and the more targets you manage to collect at once, the more of a shield you get.
Although capped at 1000 dmg, its good for baron steals as well :)

Its about getting AP and CDR as soon as possible.
An Amplifying tome into a Mejai's soul stealer is the opener, you want to start building stacks via kills (hopefully) or assists if necessary.
Follow this up with level 1 boots, and a Kage's... We are looking to build this into a DFG, rather than to use the gp10 aspect of it.
Upgrade the boots to Sorcerer's shoes, and start looking for the Rabaddons.
The last two slots are up to you - spellvamp, utility, magic pen, its up to you.
Void Staff, Rylai's, Abyssal Scepter, Will of the Ancients, or even Lich Bane.

Despite being a bit squishy, eve has good poke early game, and being able to spam q while running often turns the chase the other way.

Health is the main concern, so use pots when you need to. For mana, just stop attacking, the regen is good, and missing out on the xp not so much.

Remember to go 'missing' for a while - be it a gank or just faking it. Push bot back to tower, teleport top and gank, then run to mid. Mobility will translate well in the early game, and should give enough of a lane advantage to split push and avoid getting zoned by tanks before you finish your Rabaddons.

Try and get your Q to hit the enemy when ever possible - through the minions, keep up the harass. And when clearing a wave, remember to POSITION yourself so that Hate Spike deals as much damage as possible. Your ability to do this will set you apart from other eve players.

The old Eve was practically useless in a team fight, but the new AP Eve is much better. You have two (conflicting) responsibilities.

The first traditional one is to clean up. Chase them down quickly, and teach them to NEVER be alone.

The second is to pump out damage. Wait - Eve, pump out damage? With your AP and CDR, you are a magic damage factory, stick just behind the tank and QQQ.

Eve's passive is what its all about. Mana regen (and a free 'ss top' call) while NOT missing out on XP, useful.
Sneaking up behind a champion, undeteted by minions and sight wards, turning a 50-50 engagement into a free kill, useful.
Being able to keep an eye on a couple of champions waiting for them to separate, useful.
While stealthed, being able to ignore minion unit collisions, useful.