This is a build done by playing master yi and it's quite similar, but, there's no strange way in this... both of them can be potentially carry, better yi, but evelynn has the stealth!

Now, let me explain how to play evelynn with this build in the first moment of the game.
One word: Defensive
Play more defensive as possible, obviously if you see that your teammate is aggressive and you can be so do it... but, if it's difficult the enemy you have in front so be defensive...
Kill creeps on creeps, pay attention to any movement of the enemy and when they go under turret go in brushes and activate the stealth, as fast as possible, and stun one of the enemy, the one more near the turret... quitely idiot idea but even useful.

In the midgame, if you weren't killed all time you maybe have done some kills... depends on what you've bought from the shop but if you have phantom dancer you're in a way all in an uphill battle, you're now fast, as movement even as attack... and remember to use the final every time you have to run away from a battle or you need to gank someone. Fortunately the final has a quite small cooldown.
Phage is the most important equip you need in midgame till the end! you can have the phantom dancer but if the enemy runs away you cannot kill him! Or you go and do some jungle to have the red buff or you need absolutely that item! I've not put the frozen mallet, but if you have all equip and the game still goes on you can obviously buy it :)
For the armor penetration, if you see that the enemy has a tank with a lot of armor buy first the black cleaver or if you cannot buy it so buy the brutalizer, but when you can change so sell the brutalizer and buy the clever :)

As you can see this build is quite difficult to do, some items that costs much, but it's all logic... In the late game you will surely have a good number of kills and/or assists, so you're surely feeded. So, you will be able to push the turrets when you want, maybe when your team is fighting against the other you go in a lane and kill all mobs fast, go to the turret and destroy it... than the hinibitor, than... game won :)
One thing that I've forgot to say is that you need absolutely to see every time the map to go in other lanes than yours to gank someone and change the way of the game!

I will surely appreciate some suggestions about my build, so don't be afraid to tell me!
Only one thing: be smart and nice :)

The summoner spells are Teleport wich allows you to gank in other lanes even while stealted and Exhaust until you have phage of if someone is distant.

The skill sequence that I've selected prefers to bring up fast the stealth and the 3rd spell because the stealth have a duration higher that is really really useful and the 3rd spell do much more damage, the first is useful only for the creeps and when you have right equipments you don't need to use it because you will kill the mobs with one attack...

When the build is complete you will kill in 3 attack the 2 super-super creeps, the 2 with maces!
Also minions will not do some damage to you because you will steal so much life that you gain more than loss... about 630 to 700 damage to minions when you do a crit........... and you do a lot of crits!!

Pros: fast attack
fast movement
fast leveling
Cons: Low life
Low resistence and armor
Never, and NEVER go in a teamfight! you will be surely dead!
For you the cons are too problematic? think about the idea that no one have to see you until you kill him behind... and when you've killed someone you will return invisible to enemies!