-Summoner Spells- I take smite and ignite as summoner spells. Smite allows you to get level 2 before lanes even start, allows you to get blue buff around level 5 or 6 (which you NEED!), allows you to smite the tank minion every 3rd wave for a good amount of gold, and also allows you to gank their buffs/dragon/ and even possibly baron. Ignite is a good finishing off tool. Nuff said.
-Levels 1 Through 6- This is going to be your toughest phase. Make it through this with a decent amount of gold and around the same level as everyone else and you should be golden the rest of the game. I prefer to play Evelynn passive until i get level 6 and have my mejais. I always get hate spike first and always head to the small golem camp for an easy level 2 right before lanes start. Smite one golem and spam hate spike until both are dead. Put your second point into shadow walk. This will leave you about at half HP, so play super passive and turret hug if needed. Remember to smite the tank minion in every third wave for some good gold. Third point should go into hate spike and the fourth into shadow walk (since they severly shortened the time of invisibility). If low HP shadow walk and get close to the minion wave so your not losing experience. This is a good passive tool that allows you to stay alive in your lane and gain xp when the minion wave is away from your turret. 5th point in ravage and 6th in ult. Your ultimate is beast. Even though they shortened the time of Malice and Spite you still recover health after a kill or assist. This allows you to turret dive your opponent. I dont go off ganking until i have my mejais soulstealer. You want to start stacking this ASAP. It helps with your mid game ALOT.
-Mid Game (Levels 7 through about 14)- This is Eve's primetime. Feel free to set up and initiate ganks. Start stacking your soulstealer. Feel free to Shadow Walk around the map and through thier jungle to scout some straggling opponents. Once you've spotted one, ping your team over to help you out unless you are confident you can take them down by yourself in which by all means do. 4 points in shadow walk is sufficient enough for mid game. 40 seconds is more than enough time to scout around.
-Late Game (Levels 14+)- This is when you should have an enormous amount of AP. After your Deathcap and around 10 stacks on your mejais you should have around 500-550 AP. This allows you to nuke the squishy/mid health champions. Pick on them, make them afraid of you. Your final item should bring you to just around 800AP. And you will no doubt win the game.
-Oracles- Most people think that once the other team has oracles, Eve is useless. This is a common misconception. Eve can use bush cover just like any other champion. Oracles cannot see through bushes. Tell your team to focus the champion with the oracle in the next teamfight. Going behind a champion with an oracle is also a good strategy. It allows you the opportunity to get the stun before they can run back to their turret for protection.

-Dorans Ring- This will help you in your laning phase. You definately need the extra health, mana regen, and AP.

-Boots of Speed- Good to get early so you can avoid getting ganked

-Amplifying Tome- Get this with Boots of Speed on first return. Stacking AP early is what you need since you dont do alot of damage early

-Mejais Soulstealer- Getting this ASAP is essential. The stacks will make Eve out-damage pretty much any opponent early/mid game.

-Sorcerors Shoes- Every champion has magic resist. This is basically Speed Boots 2 with 20AP

-Sheen- This should be the first item you build toward your Lich Bane, You need the mana

-Lich Bane- Eve's Best item. Every Eve should have a Lich Bane. This item allows Eve to do an additional amount of her AP in damage after every ability she uses. Since hate spike is a spammable ability, Eve can do massive amounts of damage while hate spiking/auto attacking. Late game when you have around 800AP, you can ravage for around 1200, and then basic attack for another 800 damage. thats about 2k damage in less than a second.

-Rabadons Deathcap- This is the item you need to increase your AP substantially. When you get it your AP should rise from around 250 to 500. Massive AP ftw.

-Giants Belt- I like to get this after my Deathcap even though im not building my Rylais yet. Eve is a squishy character, and sometimes you just cant get away after picking off the squishy. This helps with her survivability alot and helps you keep those mejai stacks.

-Nashors Tooth- The attack speed coupled with your ultimate makes you kick like crazy. Adds to your AP, DPS, and not to mention that 25% cooldown reduction on your abilities which is like a cherry on the top.

-Rylais Crystal Sceptor- Last item i build is a Rylais. Use the giants belt you purchased from before to build it. Adds AP and allows your enemy to not escape.

The Nerf during the last update severly effects Eve's early game. Here are some problems I have encountered and some solutions on how to overcome them.

1. Lower Base Damage- This makes it alot harder for Eve to jungle. I have stopped jungling Eve because of it. To compensate for this loss of damage i build straight AP runes. This starts me out with 35AP. This coupled with the 15AP given to me by Dorans Ring helps her early game damage output.

2. ShadowWalk Stealth Time Decreased- This sucks because its how i used to get my exp early game. To compensate for this I now put 2 points into shadow walk early and a 3rd point into it at level 7 after i get my ultimate.

3. Less Starting Mana + Greater cost for hate spike- This one really hurts. The Dorans ring recharges a little bit of mana but not a sufficent amount to keep you in lane. I dont spam hate spike anymore because they upgraded the cost and it eats through mana. I will only use hate spike now to last hit minions. Also, the smite allows you to get the blue buff early.