As an Evelynn you are living for the mid/end game, while making sure that the enemy team doesn't live. With the runes, build and the slight insight to some of the junglers you'll be facing, you should be able to dominate a whole lot of your games.
You have disgustingly high damage, you'll find yourself manically laughing just the way she does, when you're one shotting squishy targets and peeling the armour from tanks.

As it stands for me at the minute, I've got just under 330K Mastery Points and a 63.2% winrate in soloqueue with Evelynn. I know her kit inside and out, I stick to the build I have provided, every single game in the same order. I have a system, that system works and I stick to it.

The Runes:

Electrocute: Helps with bursting your targets in the early/mid game.

Sudden Impact: Synergises well with Demon Shade post level 6. The bonus damage from leaving stealth is great. Also your E has a slight dash to reach a target, which also synergises with Sudden Impact.

Eyeball collection: Killing or assisting in kills grants you more AP. More AP? Sounsd like more FUN.

Ravenous Hunter:Make sure that you get stacks on targets you know that'll be tanky in the late game, as soon as possible. Get the Garen top, take the Leona Support. They'll be a pain in the butt later, so take those stacks and use their essence to keep you healthy when you're clearing the jungle.

Absolute Focus: Gaining adaptive force above 70% HP. What can I say? It is bonus damage. The main thing you want, is to instakill a target and leave them looking at a black and white screen wondering why they just got hit by Evelynn, the 16-wheeler truck.

Gathering Storm: We don't take Water Walking or anything of the sort, you're a BURST assassin. The longer the game goes on, the deadlier you become.

The Items:

Your full build (in this order should be): Red smite with blue enchant, Sorc Shoes, Lich Bane, Void Staff, Rabadon's Deathcap and Spellbinder.
I know, red smite. But this is MUCH more helpful than having blue smite if you're in a pinch, not having some early damage, the burn from red smite + Red Buff can still help you to finish off opponents. The only benefit you'll get from blue smite is slowing your opponent. With Spellbinder and Lichbane giving you 17% more Movement speed however, you really won't need it provided you use that to your advantage and you position yourself properly.

After you've gotten your red smite and boots, this is the build to go and this is the order to buy these items in.

Lich Bane: Just a standard buy for Evelynn, synergises extremely well with your E as it'll just empower an already empowered auto attack. You'll burst your target, squishy or tanky.

Void Staff: Your W has magic resist shred, Void Staff will add another 40% onto that. Evelynn is very good at taking down squishy targets. Though with the bonus MR shredding, you can quite literally kill squishies with just your E (at full build) and you can battle tanks too.

Rabadon's Deathcap: Who doesn't love more AP?

Spellbinder: Some targets may require a little more love. The Spellbinder will store a stack (up to 100 times) every time that you cast a spell, or one is cast near you. With your Q, that's 4 stacks straight up. Stacking this is no issue. Giving you 50% decaying movement speed, on top of the 17% you're getting from Lich Bane and Spellbinder, you'll be pretty swift. The bonus AP and movement speed given from activating Spellbinder can help you chase down or safely escape a target should you find yourself in the need to do so. However I mainly use it to burst tanks even quicker.

How do we be effective as Evelynn? You're an unseen and unstoppable force if you play her properly and utilise her kit as much as you can. So let's go through each step of the game.

Early: We're level 1 and starting out. You're not tanky and you don't do a whole lot of damage with your Q as of yet. Always start botlane, you'll have more help with your camp and a number of supports will be able to heal/shield you for a healthier clear.
Red Buff - Raptors - Blue Buff
or
Blue Buff - Wolves - Red Buff

Now that you have both of your buffs, let's take a look at the map. Depending on how long that took, the Skuttle Crab should be showing either a gray indicator or a yellow one to say it'll spawn soon. I don't need to tell you that the vision and gold that these babies give are highly useful, you know that already. However the enemy jungler might have a different idea. If you're not health enough to try and poke the enemy away from a Skuttle Crab, think about where they are. A lot of junglers tend to start at their bot side and work upward. If you're going up against Ekko for example, and you started Blue, you've just taken your Red buff and you're in the topside jungle. Skuttle is showing the grey indicator. Take your Raptors, run to the bottom river and secure that Skuttle instead, just trade. Stay healthy and don't lose your buffs. Pick your fights.

Demon Shade, your wonderful passive, won't heal you too much at this stage, but combining it with your refillable pot is a great way to stay healthy. We need to stay healthy and in order to do that, we must kill enemy champions. Ravenous Hunter won't stack itself. Take a look at the map, see which of your laners are pushing or getting pushed. See if any of your enemies are less mobile than your laner and are easy to lane your charm on. Get some early stacks where you can and continue to make map pressure.

Pre 6, ganking can be a bit of a pain. You're not awfully fast, though in the top and bottom lanes, depending on the minion wave positions, you can just stroll into one of the bushes and MAYBE use the alcove to get around. Wait for the enemy to start pushing a little more then spam ping a target, mark them with your W and go to town.
Post 6 however, its more or less the same. Enemy laners tend to Control Ward the river, thinking that you can only come from there. Realistically you can be in their lane and behind them without them even knowing. Being behind them is where you want to be before hitting W.
Your enemy, 90% of the time, will run toward you to help you out. The other 10%, they'll stay right next to their team if any are around. As you mark them, when they see the mark and hear that sound, they know that death approaches. As they run back, keep your distance until its too late. Use your Q to initiate (though your E will do more damage, your Q is usually safer vs ranged targets) and then E into the rest of your Q. Ult to finish.

Always be looking for which enemies you can get behind, if they're hard pushing their lane, continue to punish them. Make them embrace agony. Should you be destroying your jungle opponent, make sure to set them behind even further, steal their camps, ward their jungle. Give your team info on where he/she is.

Ganking midlane. There are two angles that you do not want to gank from as you'll be spotted by the tower. If the enemy has the top side, do not gank from the RIGHT side of the midlane tower. You will absolutely be spotted even if you hug the wall. If the enemy has started on the bottom side, steer clear of the LEFT side of the middle tower. Just don't go near Blue Buff side of either tower. It won't do you any good to be seen.

Even after level 6, keep using bushes to your advantage. You're not inivisble, you're using camo, this means that unlike Shaco, you will be seen if somebody is close enough to you. When you're in a bush however, provided there's no control ward in it, you can sit and wait for a target to be in E range and initiate with that, nearly taking them from 100 to 0 depending on what stage in the game you're at.

Evelynn's ultimate is an execute, there's no arguing that. Though it has multiple uses. If you're in the back of drake pit, sneaking a dragon and you kill it literally moments before the enemy mid and bot lane collapse on the pit, you can just ult over the wall to safety. It has quite a decent range and is more often than not, miles more useful than burning your flash.

Keeping objective control as the jungler is VERY important. In the current season, the dragons are especially important. Do your best to keep it warded. Try to keep pressure on your middle and bottom lanes to give them priority should you need help with taking the drake.

Naturally, killing the enemy botlane and then rotating to the dragon is a great call. However, should you find yourself needing to do the dragon alone, here are a few tips. Keep the enemy jungle warded, go for drake immediately should you see the enemy jungler in their topside jungle or toplane, hold your R and always start fighting an objective by marking with W and then using your E. The bonus damage you do, even early will be nearly 1000 damage. When you have a few items, look for dealing around 1.5-1.8K. If you're trying to shred a target, lead with that, then to end, couple that damage with your smite, you can deal 2.8K damage in under half a second.

At the time of writing this guide, I'm Silver IV. The highest I've ever been is Silver II. Though you can see (as its all public info) that I don't really play ranked. I'd get to Silver for the border then go back to normals.
However, due to my MMR being so high, I still play against Season 9 high plat/low diamond players.

At the moment, I can't think of anything to add, as it is my first guide, I'll probably come back and tidy it up should I need to at a later date. However, if you've gotten this far, thank you for reading.