This is just a build I made for my own reference, and Im publishing it for my friend because I'm a tard and don't want to screen shot it and upload it. (my friend, you know who you are) :D

My AD and AP build are both up to date as of now, I have yet to test the AD build in PvP, but in practice, this build wrecks. Feel free to comment on my guide, try out the builds and see for yourself how scary Evelynn can be. (:
I thank you for reading my guide!

P.S.

Today I was able to solo Baron with this AD build, not even including the Last Whisper. Just make sure the Bloodthirster is stacked, and you win. :D

Pros:

+Nuke almost anybody
+Lots of speed
+fairly quick jungler
+lots of damage mid-late game
+(AP) Is a huge danger to tanks or squishies, with the %health taken with both her Ult and her Deathfire, she's a very deadly assassin. Health is no object when faced with an AP Evelynn. (:

Cons:

-Eve very squishy squish
-(AP) Easily countered by Magic Resist
-Stealth is detected within a fair distance
-(AD) Countered by armor

The items that I have chosen for AD are based off of generic AD builds used for Caitlyn, Ashe, Sivir, and so on. For Evelynn you definitely need that gp5 boost, unless you're very good at laning and harassing with Eve's Q. With the Attack Speed boost that her E gives her, Evelynn does not need much attack speed by itself, so I tried to focus more on Damage, as opposed to the speed she attacked.

Additional choices:

Instead of the Berseker's Greaves, you could switch them out for another Phantom Dancer( or Blood Thirster, or I would recommend a defense item such as Guardians Angel or some type of armor/Magic Resist Item). With the Trinity Force and the first Phantom Dancer, you should have quite a lot of movement speed.
This AD build is still under construction. :P

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The items that I have chosen for AP are mostly what I do for jungling, most the time I will start Amplifying Tome and 1 Health pot, and have a really strong leash on the blue Golem. Next I move on to Kages Lucky Pick mostly because it gives you that extra gp5, and builds into Deathfire Grasp Later. The rest I move into situationally; Rylais if I need health, Sheen, (builds into Liche Bane) then Rabadons if I want lots of burst. The 6th item to your build is completely up to you, I would recommend a Void Staff(or Abyssal Scepter) only if they have high MR.

Generic Rune build for AP Nukes. Lots of AP, and armor for the jungling portion of the games. (: That extra AP boost early game helps a lot with jungling, so that you do not need to get a wriggles. I have seen many people that jungle as Evelynn, move into a wriggles for easy jungling, even though they are going AP. Personally, I think that is a waste of time and money, you should be focusing on ganks, or counter jungling, only really watching your buffs. If you pull off ganks correctly, you'll be getting your AP items, and won't need that Wriggles.

Warning:I am no professional jungler, this is just the route that I take when I do jungle.

This build isn't specifically for jungling, it's just a generic AP glass cannon build. With this new Evelynn Rework, she can now clear the jungle with no problem at all, whether she has cloth 5, or Amp 1. I normally start at the Blue Golem and have a very strong leash, to at least half health, so I don't take any damage. That's when I move to Wolves, then Wraiths, double Golems. At this point I probably have to back, before I get Red.
If you wish to know the details of jungling, then you will have to look up a jungling guide, because I don't have much information for you.

For Evelynn, I think that her skill sequence should be Q, E, Q, W. Maxing Q first, following with E, and leveling up her Ultimate at all available levels. There's not much reasoning behind it other than the fact that her Hate Spike is spammable: does a good 80-100 in it's lower levels.

There is a reason that Evelynn is not chosen often for Solo Q. What's this reason, you ask? Randoms don't want to deal with babysitting an Evelynn. That's a fact. You need a good team to get fed to the point where you're actually a threat. Or you need good communication when jungling, so you can pull off ganks well. Evelynn has never been, and won't be a good Solo Q choice because of the fact that she needs to be babysat until mid-late game. Or until she gets fed enough between gp5 items.