Hello there summoner, welcome to my guide, Tango'ing Your Way Out Of ELO-Hell FT. Evelynn..

Shadow Walk

When out of combat for 6 seconds, Evelynn enters stealth only being able to be seen by enemy champions (within 700 range) or true sight. Evelynn regenerates 1% of her maximum mana every second while stealthed.

Hate Spike

Evelynn fires a line of spikes through a nearby enemy, dealing magic damage to all enemy units in its path.

Dark Frenzy

Evelynn removes all slows affecting her, gains increased movement speed for 3 seconds and ignores unit collision. Champion kills and assists refresh Dark Frenzy's cooldown.

Ravage

Evelynn slashes a target twice, dealing magic damage with each strike. Evelynn then gains bonus attack speed for 3 seconds.

Agony's Embrace

Evelynn impales all enemy units in target 250-radius area, dealing magic damage equal to a percentage of their current health and slowing their movement speed for 2 seconds. Evelynn siphons their pain, gaining a shield that grows in strength for each enemy champion hit and lasts up to 6 seconds.

This increases your magic damage.
More health in laning phase. Yes please!
This can be great against burst champions like LeBlanc, take Greater Glyph of Ability Power if you aren't against a bursty champion.
This quintessence is pretty self-explanatory, it gives you more AP for your skills which can add to Lich Bane.

This guide is a work in progress so instead of down-voting please tell me what you didn't like about the guide in the comments. Thanks!