I mostly play her defensively in the beginning.I try to farm as much as I can with Q and when I'm low on mana I let my passive activate and give me Mana Regeneration.I don't push the lane, but because of the W and Flash it's really difficult to get ganked.Because of the recent nerf on the champion, she's no longer considered an early game champion, but a late one.When I get my ultimate, I try to gank top or bottom lane depending on which is the most pushed. Your goal on the game is to push the lane and gank. This is made easier with the help of your Passive which makes you invisible from Sight Wards. The most effective way to get a kill is to harass with Q, lowering him enough before you cast your Ultimate. It's pretty tough to get away from Evelynn due to her W. I wouldn't suggest in any way to rush Death Fire Grasp because of its recent nerf and so it will be much more useful after you've bought adequate damage items. Also, with the last changes, it's better to use your ulti first and DFG later for more damage.
Tips & Tricks
For starters, let's talk about how Evelynn's first ability works. It generally gives priority to champions instead of minions, unless you've attacked something. In that case, it will hit the target you attacked in the last seconds.
Try to set the pointer between you and you enemy on the lane if you want to harass him, or on the minions to clear waves.
When you want to recall, make sure the enemy doesn't know by waiting until your passive is ready and make sure you stay at a place where he can not see you.
If the enemy can actually see you, move to the end of the lane at the bushes before turning invisible. Now, the enemy won't be aware if you're still there, if you recalled or if you're roaming for ganks. If you keep doing that, your moves won't be expected and you'll be able to move freely on the map.
Also, you can use your ultimate as bait to reverse the flow of a fight. Let them think you're vulnerable until you get the chance to turn around and kill them.
Evelynn's counter picks is Mordekaiser, for starters, because he's a lane pusher and due to his shield he doesn't get easily harassed. Most definitely you won't win the lane over him.
Katarina is also a counter, but it's important to just play safer, get magic resist and wait for the jungler to help out.
Finally, Swain is a strong pick against Evelynn because of his CC and spell vamp and can easily kill her.
These opponents have made it hard for me so far.
From then on, Evelynn is strong against Morgana if she can bait her to waste her shield.
She's also good against Fizz because of her first ability's damage.
She works great with Amumu as an ally because of his AOE stun ultimate, combined with hers AOE damage and slow.