For items, instead of Rylai's Crystal Scepter you could go with a third Rabadon's Hat or a Mejai's Soul stealer. Go for the hat, unless you know for a fact you can get stacks without dying, or are feeling lucky. For boots, choose either Boots of Penetration or Boots of Mobility.

When playing Eve, try to work together with your lane partner for an early kill. If teamed up with poppy, stun an opponent and let Poppy stun them against a wall and beat down the target. If laning with a Zilian become invisable and have him put bombs on you, run at the enemy and stun and use ravage to lower magic resist before the bomb goes off. Eve is about ganking, and taking advantage of fleeing or weak enemies. Don't be afraid to tower dive either. Eve's ultimate will give you enough life to live after a kill. Also try to pop your ultimate before an enemy dies because it will come back right after.