Hey there, summoners! In this guide, I would like to present you my way of playing Evelynn. I've been playing eve for long, testing out all these AP(not so viable imo), AD jungling(got nerfed ;_;) builds. But this is something different. You are not a jungler. And not a laner at all. In next parts of guides I will explain your role in team.

Pros

-Extreme ganker
-High damage output late game
-Element of surprise when you come out of Shadow Walk
-Roaming often carries your team to victory
-Extremely fun to play

Cons

-Squishy
-Low damage early on
-Countered by oracles early on
-Misunderstanding with team as a roamer may lead to defeat

For runes I take:
Greater Mark of Desolation 9x - gives you needed ArPen for ur DMG
Greater Seal of Scaling Mana Regeneration or Greater Seal of Resilence - your personal choice. But I usually don't run OOM so I take Armor ones
Greater Glyph of Scaling Cooldown Reduction - CDR never hurts
Greater Quintessence of Health - makes you more durable, more scary in early ganks

For masteries, I take 21/0/9. These masteries boosts your damage output, that is not really strong early on. They make your ganks succesful :P

Items here are not something special as for AD Eve, but i'll explain my choices.
Boots and 3 HP pots - standard for this build, makes your switching lane fasters, better chaser, helps you escape and these health potions make you stay without returning loong time - MANDATORY
Sheen as second item. Boosts your damage output drastically. This passive is soooo great when looking at your Q cooldown.
Mobility Boots Mobility as a roamer is everything.
Trinity Force This. Is. GREAT. Good all around item, but also boosts your Sheen passive, gives you needed slow. THIS IS MANDATORY.
Youmuu's Ghostblade Preety good item, active is kinda ur second ult, armor penetration, some AD and crit chance with critchance from trinity makes you crit frequently.
Banshee's Veil IMO best survi item in-game. If they have lots of AD chars you can take GA here.
Infinity Edge More AD, more crit. Do I need to say more?
Guardian Angel Great survi item, armor, resist and cool passive

My skill sequence here may look a bit strange, but it's preety good on roaming eve. For first point you should take Shadow Walk. Very good while ganking their jungler or lanes. The stun is just a bonus. Next you should take Hate Spike this skill makes your Sheen proc. Of course maxing Ravage then is, as it is a good nuke, with reduction of armor. Take ult at lv 6, 11 and 16.

For summmoner spells I take Ghost and Ignite. Ghost is good for chasing/escaping. Ignite gives you some more dmg output early on, what you're lacking.

Your role in team is strange. You may be thinking "Why! for real! that's ******ed." or "OMG NUB EVE!!1111oneoneoneeleven". But no. It works. What you do is called roaming - you do not take a lane, from the WHOLE start you run on different lanes and gank your enemies. Of course after a succesful gank stay on a lane for a while, so you won't be outleveled. It works awesome because:
1. Makes your team get much kills early game with these ganks
2. Makes enemies lose lot's of gold for wards/oracles, that are lost anyways when u gank with ghost + ult combo lol
3. Makes your enemies play much more carefully, and sometimes even resign from last hitting minons, as you can be everywhere
4. Gives a solo lane in team, making your team get more exp.
Roaming takes lots of practice, you may fail at start, but this is highly rewarding, and often makes your team win lol.

Pick up Boots 3 Health Potion. If they have a jungler, you should go and gank him. Evelynn is a great jungle ruiner. Basically what you wanna do is to gank their jungler when he is low hp finishing blue/golems or anything. If they don't have a jungler, pick a lane(solo or mid imo) and gank. With help of your ally you should be able to get first blood, if not, come to the same lane again in a while. In early you want to gank as much as possible, to dominate enemy team. How much did it happened when an TOTAL early game domination made a victory? That's what are you for.

As I said earlier in the guide, after a succesful gank you should stay a while on the lane, to get some exp. Your level shouldn't be lower than duo-lane level. In mid game, their jungler should be ****ed up(of course if they have any). You should still be ganking in mid game, with your Sheen and Mobility Boots you're able to move through lanes fast and deal LOTS of damage, so now gank should not be a problem. Sometimes teamfights happen in the mid-game phase, you should not be scared of them, unlike AP Evelynn, AD is awesome in teamfights. I'll explain her role in TF's later.

Now, you should find and eliminate(lol) champions that aren't with team. In better teams, they will be all together, so you have to win teamfights :3 Control baron with wards, play safe.

Ok, what maybe you've been waiting for. AD Eve is preety good in teamfights. What should you do? First, focus their carries. With your stealth, you're able to get to them, and you destroy them. FAST. Of course sometimes you won't be able to focus them, then hit anyone but tank. Spam your Q as much as possible, as it no only deals some dmg, it activates your Trinity Force passive. Always keep your ultimate on, as it gives you high damage boost.