Evelynn is one of my favorite champions so when I looked for a guide I realized none of them fit how I liked to play Eve. So I decided to make my own.

Pros

Great speed
Amazing ganks (why she is best jungler NA)
Great Damage
Great at getting away
Great Initiator

Cons

Low Sustain in early jungle
No Great CC
Slightly squishy early game

Now you're in the jungle. What do you do? There are three viable jungle paths for Evelynn, depending on whether or not your teammates will help you and which side you are on.
If your teammates help you.

The Path On Purple Side

Go for a smiteless Blue Buff then head straight for your red buff to make sure the other jungler doesn't steal it and smite it. Then do wraith then wolves and recall. Buy the first back items. Then start ganking lanes that are in trouble.

The Path On Purple Side
Start with a smiteless Red Buff then go to blue and use smite take wolves then wraiths then recall and buy the the first recall items. Then you can gank.

If your teammates are ****s and wont help you

Always take Red Buff by yourself and try to take wolves and wraiths then recall hopefully your smite will be back up and you can take Blue Buff then gank the lane thats in trouble.

As I listed in the pros section the ganking is brutal because of your invisibility. Your ult is amazing for ganking because it slows them by 30% and grows. Try to initiate with your ult.

Try to hit as many people as you can with your ult either as an initiation or under tower then W to catch up to them and spam Q at them for damage.