Not Updated For Current Season
First you must know what can evelyn do...
-Basically, you need to pick your fights carefully. You need to know if you can kill someone and if you can't and this only comes with practice. This is the most important skill when playing Eve and takes a long time to learn as there are so many variables. Learning when to go for the kill is what separates the good players from the great.
Note:I know you probably have a lot of preconceptions about Evelynn and you have probably looked at the build and seen things like smite and thought "noob" but just trust me :)
Okay, I have been playing Eve for a loooong time now and as she is free this week (well, the week I wrote this) and I have seen a lot of people trying her out - and failing... badly. Eve is a very hard champ to understand and therefore quite hard to play. Also there is no guide that I can find on Mobafire that is actually correct, so I will attempt to write a definitive one. I'm not saying it's perfect, but if you follow this and PRACTICE it I guarantee you will do well.
To reiterate, there are a lot of ways to build Eve and succeed and this build can definitely be tweaked a bit to suit your style. What I aim to provide here is a general formula that provides consistent favourable games. The fun comes when you perfect the basics and start to make this build your own.
So what can Eve bring to a team? I have been called Kill Stealing noob, Paper champ etc but actually, Eve can do a lot for a team. Yes she kind of does kill steal early-mid game. But she needs to and your team should understand that and work with you, not against you. Besides, what is better, a person getting away and your team-mate getting no kill or them getting an assist? Eve basically GUARANTEES kills. She is like Garen or Karthus's Ulti, only everywhere and all the time. Not to mention, she is a great scout, Really good at taking turrets down with her ulti and a really good counter to junglers. I can't tell you how many times I've waited by ww or someone only to smite steal blue and write /all Thanks mate! Even done it with Baron once which pretty much won the match.
Okay, now you should be pretty unstoppable. As team fights start to happen you should think of yourself like a yoyo. Bouncing in and out of fights to finish people off. Don't be afraid to run into a group of 3 or 4 people to kill one person on 20% and run off again with your ulti. Remember your ulti also recovers your HP when you get a kill. This also makes it an amazing tower diving ability. Champs on low health who have retreated behind their tower to recall are free kills for you.
If someone gets oracle then tell your team to focus them and play a bit more carefully. Just be aware and you shouldn't get into any sticky situations. It's generally a good idea to assume that everyone can always see you. Try feinting when in stealth to see how people react - it can save your life. Keep making them waste 400g on oracle and more on wards and watch as your team gets a significant item advantage.
After you have Lichbane and possibly a blue elixir, sell your stone and buy fiendish codex. You should have made the money you paid for your stone back through its G per 5. Then get your Rylais, Rabadon's and lastly upgrade your codex to Nashors Tooth. By this point you should be pretty fearless. Depending on how fast you got your build you will easily be able to go up to a squishy stood next to their tower. Kill them in like 1 second and walk away again. THAT is when people start saying things like "no skill Eve" - and by then it pretty much is, you can smash your head on the keyboard and rape faces. What they don't think about is all the effort it takes to become unstoppable like that.