Evelynn is a fun champion to play, but she is tricky. I like to play Evelynn for the fact she is filled with strategy, from how to build her, to how to play her.
She is not a pick up and go champion! If you want a super easy champion to play, go play Warwick or something. Once you get the hang of her, she is extremely easy and fun to play.

Items are important. The route you're going is important too. Are you going Tanky Evelynn? Are you going AD Evelynn?

Tanky Evelynn

When going Tanky Evelynn, you finishing your Ranger Trailblazer first is important whether being AD or Tank Eve.
If your team has 2 tanks, don't be Tank Eve, unless your tank is trolling, feeding, or is all around bad. Having 1 or no Tank is when Tank Eve comes in hand.
Finish Trailblazer, then start building Frozen Heart, after you get some boots. Doing this will build armor while cooldown reduction. Then do Omen, where this has more armor, and health. After you get those items, go and start building AD.

AD Evelynn

With AD Evelynn, the rule applies, you need a Tank? Be the Tank. It's simple. What's important, kills or winning a game? If you have the Tanks you need, just go AD. Build your Trailblazer, then Boots of Mobility. After that, start building into Bilgewater Cross. This does build into Blade of the Ruin King, but you wanna stop and see what's going on at this point. You want Tank or AD. Make a choice, if AD build into Blade of the Ruin King. Trinity Force is an option. Trinity Force has some AP in it, but boosting some AP won't hurt.

This chapter will focus on the strategies you follow when playing her.

Jungling

Start off with Gromp, the big frog. When he is half health, smite. Yes everyone says do it at the last second. No don't with Gromp. When you smite Gromp, big frog, he will give you the poison ability. You will finish him faster, if you need health, W then run to Red Buff, if not use a health pot, then go after Blue Buff. I believe Red is more important than Blue. When playing Evelynn, you will never run out of mana, unless you go AP on her, then you might.
Do not be afraid to use W when running in the Jungle. No mana cost movement speed. Dream come true.

Creeping

This is the real fun part. Whenever you need to gank, have your teammates bring the enemies up to the turret. Then sneak behind them, hit W then R, then start smashing E and Q on them. R will slow them down as they try to run, then using E will knock their health down, then smashing Q, Q is on a barely 2 second cooldown. Smash that Q till your keyboard is a pancake.
Before you gank, make sure to use your Red Trinket to check for wards. They will see you, and everything will be screwed.
When going into a team fight, save W. You go in and start comboing and then bam! Everyone is dead but you, what do you do? W isn't an option when running away. You have an advantage with having a movement speed boost. Use it wisely. 12 seconds is a long time.

Tanky or AD? Your choice, choose wisely. Need a Tank, be the Tank, if not go AD.
Skills go Q, W, then E. Max Q, then E then W. Update your R whenever you can.
When Jungling go Gromp, smite half way, then finish him. Go either Red or Blue, whichever you need more. Use that W when running the Jungling.
When Ganking, go behind, if possible, then W, R, E, and Q. Do not be afraid to use that Q when attacking.
Team Fight, save the W, just in case for an escape plan.