How do I jungle as Evelynn?

As Evelynn, the easiest way to jungle would be:

Red Buff -> Krugs -> Raptors -> Scuttle

This should get you to level 3 by the time you go to take Scuttle. Level 2 Evelynn doesn't have a strong Scuttle contest so just try to get level 3 beforehand.

Another key thing to note is that you want to land your Hate Spike (Q) as much as possible as it will reset on jungle camps. Evelynn's aa (autoattacks) are not very strong so avoid aa on jungle camps. Instead, just kinda run around and keep Qing until the camp is dead. Evelynn will take less damage this way and will make it possible to back at 1k+ gold each time.

Try to gank as Evelynn when there is a high possibility of copping a kill or assist. If there is not a high possibility, just focus on clearing the jungle and hitting level 6 as fast as possible. Level 6 Evelynn's ganks are much more useful as her ult and cloaking make her more formidable.

Last Caress is a versatile ultimate ability. "Evelynn briefly goes untargetable and decimates the area in front of her before warping backwards a long distance."
The AOE possibility is huge and if capitalized on, you could do massive damage for your team.
It can be used not only as an execute, but an escape mechanism. If in a difficult position, use the ult to escape. An alive Evelynn is a helpful Evelynn.
Her ultimate is also very helpful as a gap closer. If an enemy is running or just out of reach, using Last Caress "backwards" to get closer is a possible tactic. Just be aware that you may be ulting into the enemy team if you are not careful, because once you use it your escape mechanism is also gone.