W - Shadow Walk  

As it says, you go invis, if you attack a hero when you are in invis, they are stuned for abit. TIP: if you can always try to stun them with their backs to you, so you can use E and get the minus armor bonus

E - Ravage  

This is your semi nuke spell. It does decent damage and takes down their armor if you are behind them.

R - Malice and Spite
(ULT)  - now an activateable skill, increases attack speed and movement speed. Also once you kill a hero you get a good amount of health.

Skill build: switch between invis and hatespikes. 1 point in ravage and ulti when possible

Key Items:

1. Movement 3 boots
2. Soulstealer
3. Sheen - Then onto Lich Bane

Starting items: 

Boots 1 (I refer to my boots as boots 1, 2, and 3. It relates to the movement 1, 2, or 3 boots, not the number of boots you have. Boots dont' stack).

3 health potions

Early Game: Out of base you can really select any skill you want, but Q (Hate spike) is usually the best to start off with. It helps with your last hitting and thats extremely important. You can either solo lane or share a lane, but usually for this build, its preferable to solo lane. In a solo lane you can last hit better and gain more gold, however the downside is that your push is horrible. Most likely your opponent will have lane dominance and your tower will be the first to fall. Therefore, its best to take either top or bottom, as its the longer lanes, so you don't lose the middle lane. If you share a lane, try to get as much last hitting on creeps as possible and get your partner to help out with the kill once you are of adequate level. 

Champions to avoid: 

Zilean:

He is nearly impossible to kill solo early game. His starting skills are just too good, and you can't dish out enough damage before he runs away. Also his bomb is a great harasser, and you will find yourself quickly out of health when you are trying to last hit creeps. If you have a Zilean against you in a lane, ask to switch.  A half decent Zilean will shut you down.

Range Dps:

Range dps heroes are pretty hard to lane against, though not as bad as Zilean. Just be on your toes and try not to take too much of their harasser damage, or get yourself killed.

It is very important for you to survive early game, try not to get killed/ganked early game or you will lose out on valuable exp and gold. I'd say 1 death max. If you are having trouble staying alive in your lane, ask for help or switch lanes. Use your potions wisely and try to make them last until level 5 or 6, but of course its more important to stay alive. Use grass wisely, to prepare for later kills.

Once you hit level 5, its time to consider killing one of the champions you are laning against. If you are at full (health and mana) give a shot against a full hp champ, if not, only try to take them when they are at half health and you have enough mana for W, E, and a few Q's. Try to attack them near your tower so they have farther to run to get to safety. Your attacking scheme would be to back out of the enemy's and their minions' LOS, go invis, walk behind them, hit/stun, E, and then start autoattack/chase while mashing Q. Try to position yourself between them and their most obvious escape route and try to have only the champion in the range of Q so that it hits them every time. By level 5, if you haven't died, you will probably still only have the boots, if you have died or have gone back and bought, have the Movement 3 boots and maybe a Amplifying Tome. If you get a kill hooray, if not and your are oom or low on hp, go back and heal, most likely they are low also and will leave the lane. PS DO NOT chase them past their tower. You will die if you do, and the key is to not die with Evelyn (Unless you have a flash and they are very low on hp, if you can flash past them as they run past their tower and hit/hate spike, then do so, but get out right away through the jungle afterwards). If you are against a dual lane, pick the squishier target (Annie before Tibbers and no stun up, Casters without stuns, Ranges without good escapeability - Ashe).

Mid Game:

Once you get your first few skills and farm up to a Soulstealer/Sheen (I recommend soulstealer before Sheen, but you can get either one first). Start moving onto other lanes to gank. Pick on targets that are squishy or have low health. Move on them when they are pushing past the mid point into your side of the river near your tower, use the same attack pattern. Once you get your ult, make sure you use it before going invis, or it will knock you out. Continuously move between lanes and gank/kill as much as possible, you want them to fear your evelyn not not wander far from their towers. Now pick a decent lane and push with your teammates. Take down the towers 1 by one, pop your ult when attack towers so you can take it down faster. If an enemy gets near to defend, and you have good health and mana, step away from the tower, try to lose yourself in the minions and champions, go invis, then attack that hero. If its a tank, avoid at all costs, its not worth your time or mana, you won't kill them solo. By the time you start pushing have both soulstealer and sheen, and be working towards your lich bane.

Late Game:

Once this stage is reached, go buck wild. Position yourself in favorable locations and ambush lone champions. Its possible for Evelyn to suvive against 2-3 champions if you play it smart. Pick of stragglers or squishy targets (By this time you should be able to kill them with ~4 hits), then pop your ult and run away. When you are in the clear go invis. And if there is a squishy target chasing you, double back wait for them to push past you, and there is a good amount of room between them and help, then kill again. If you have done your job effectively, the opposing team should be a few levels under your team, push as much as possible when they have less champions after you ganked 1-2 of them and take 1 inhibitor as fast as possible. You don't want to be fighting level 18s, its best to win before the game gets to that level.

Good end game items to consider are:

Soulstealer, Lichbane, Wit's End, Recurve Bow, Black Cleaver