Not Updated For Current Season
This is my favorite build for Eve.
There are a few items that you can exchange depending on the circumstance.
Sheen will be what makes you completely dominate early game especially if you stealth then open with Ravage.
Boots are next on the list. Eve needs to be able to stay in range for Hate Spike if she is pursuing people. Every other Hate Spike will proc Sheen. She also needs to have them to be able to maneuver around the map for maximum "run away, stealth back, stun, kill"
Mejai's Soulstealer can be replaced with Rod of Ages if you aren't doing so hot or Void staff if you are up against a lot of MR.
Zhonya's is a NECESSITY it will push you past 800 some games.
Guinsoo's Rageblade. This item IMO is amazing for Eve. She does do a lot of melee attacking along with her abilities and this stacks so that you are getting attack speed and ability power from the stacks. The speed lets you waste the least amount of time between Lich Bane procs as possible.
Early Game: Early game is really simple. Stealth. Stun. Keep them out of range of their minions. You can keep them out of range of XP if you get paired against the right group. You can definitely stop them from last hitting minions. If at all possible, lane with Shen or Rammus. Someone with a taunt works amazingly well with Eve.
Mid Game: After the first tower is down. Or when you hit level 4 or 5, whichever is comfortable in the situation, start ganking. Disappear for a few minutes, make everyone play back. They will eventually have to come out, especially if you can get your team to work with you and let their minions push to your turret before killing them. When they do extend out a bit, be there waiting with a Ravage out of stealth and Hate Spike your little heart out.
Late Game: If you are fed enough, late game is a breeze. Care for oracles. They do not shut you down by no means, just pick the right time to come into the team fight and just bounce in and kill a squishy, then bounce right back out. Depending on your gear you should just about 2 shot them. Have your team Focus Fire the one with the oracle unless it is a tank, then just chase him down and remove it. Even if they keep getting them, that's 400 gold per death they are wasting.