Evelynn is a fast, difficult, and mostly fun champion to play. She can be the problem of your team or the way you won, it all depends on how the player is.

In early game laning phase you want to angle your "Hate Spike" to hit as many creeps, and if you can hit your enemy. Don't be afraid to push, seeing how you got "Dark Frenzy" at level 2 if you need to retreat shouldn't be hard. Having "Ravage" will help you get first blood, or a higher creep score.

If you have bought all your items, you are ready to start pwning so nooblets. Dealing un-godly amounts of damage and being able to run from danger. Evelynn is one of the few, "Perfect Champions"

Pros: She can hit hard, and walk of with no damage done to her.
She can use her passive to gank the hell out of other lanes.
Its hard for the enemy mid to keep tabs on Evelynn because of her passive.
Cons: Evelynn cant take much damage. (if you wanted to you can get GA)
She isn't much use unless the numbers of allies in a fight match with the numbers of enemys.