IF YOU FOLLOW THIS GUIDE YOU WILL NOT MAGICALLY BE ABLE TO GET 20/0/0 LIKE YOU WOULD FOLLOWING AN ASHE GUIDE, EVE TAKES PRACTICE, SKILL, COORDINATION AND TAKING ADVANTAGE OF YOUR ENEMIES MISTAKES TO DO WELL WITH.

Pros

Has a stun
If played correctly can carry a team to victory
Amazing anti-carry
Jungles decently
Sets up kills well
in built ghost with her ult
very strong mid game

Cons

If your team doesnt call vw's/oracles it can lead to you dying
very weak early game

This is how I play the amazing assassin that is Evelynn, i've watched many try and fail with multiple other builds, and this seems to be what works for me.

Reds: Since you aren't getting Sorcerer's Shoes since you will be roaming, you needs these marks for the MPen.

Yellows: Considering how magic heavy most teams are, these are normally your best choice, because when you're not stealthed your vulnerable. Armor Yellows are also great, and can be used for jungling.

Blues: More Ability Power giving you a stronger early game since you are starting with boots.

Quints: Provide a decent amount of AP late game, and become better then flat ap quints at around ~lvl 11 which is when you will be most dominant.

The Good:

Flash
Ignite
Teleport
Exhaust

using any of these summoners on Evelynn are fine, Flash is almost a must though, I use flash/ignite because at ~lvl 3 I go to mid for a gank if they're over extending and like to secure first blood.

Typical 9/0/21 build picking up Blink of an Eye for the reduced CD on it.

Boots of Mobility / Lich Bane / Rabadons Deathcap is your core build, after this you will be hitting like a truck with your E and doing decent damage with your Q aswell.

Get a Rylai's if you don't have any good slows on your team.

Banshee's is almost essential for Evelynn aswell, saving your life from the likes of karthus's ult and that max ranged nid spear that happened to hit you because you were moving between 2 creeps.

Zhonya's Hourglass is also an amazing choice for Evelynn, the reason of this is because if you pop it straight after you use your W you're guaranteed to get back into stealth, allowing for the safe getaway after your noob caitlyn focused the tank in the team fight causing you all to die.

you should only be initiating someone stealthed, doing your nuke and then going back to restealth, your typical skill order will go like this;

W(Stealth)
E(Ravage(from behind ofcourse))
R(helps you nuke)
Q Spam
did you fail to kill her in that 4-5 seconds and shes still 50%+ HP?
Go back to step one or stop focusing the tank.

Not going to be going into detail with this one, if you are confident, try it out. start at blue with a leash, make sure you ignite the blue golem and hes the closest to you or else your Q wont be hitting him for full damage.