This build is meant for intelligent players who value the team. Some Basic Tips Are:

-Sometimes early-game ganks are worth it. Sometimes they're not.
-Watch the level of your opponents and your teammates to know what they're capable of.
-Help during laning and use your (hate spike or ravage) + ult out of stealth to stun opponents and save the lives of your teammates when they're losing a fight.
-Don't be constrained by a single lane. Evelynn's stealth is a huge advantage in getting involved in fights across the map. Like Karthus, she should be able to kill unprotected low-health characters almost anywhere on the map.
-Although it is fun to kill, make sure you watch Eve's level, as it tends to drop below others' levels when you focus primarily on getting kills.

Evelynn isn't a hard character to use. Just be smart and you'll do alright. She is unfortunately a highly situational character. If the opposing team is getting fed by your team early, its doubtful that you will be able to nuke with her. Thus, picking the exact build order for Evelynn will depend on the progression of your match. That said, she is a very useful character for filling certain roles, which I will get into.

During the laning phase, Evelynn is squishy, but she is still useful for:
-keeping opponents from getting experience during the laning phase as the longer you can stay stealthed nearby opponents, and keep them far from the creeps, the stronger a level advantage you can maintain in a lane.
-getting some quick kills. (This should not be the primary objective, but only secondary to securing a level advantage during laning.)
-moving in and out of lanes to hold a numerical advantage in multiple lanes because the opponent does not know where you are.
-Using the auto-attack ravage combo is critical to dropping the opponents hp enough to launch a killing blow later.

The opener is important. Giving your team that level advantage by using Eve as a mobile threat is the key reason that you should start your build with the below items.

If you learn one thing from this guide: it is to adapt. The core items most important to this form of Evelynn are movement boots, 3 hpots, mejai and sheen.

http://www.mobafire.com/league-of-legends/item-purchase-planner#&item=1-27-27-27

Depending on the timing of your kills, the next item to get is one of these three items.

http://www.mobafire.com/league-of-legends/item-purchase-planner#&item=86-50-55

All three of these items are central to Evelynn's core AP build.

The Boots of Mobility allow Evelynn to catch fleeing opponents while stealthed and drop the stun on them.
Sheen vastly increases Evelynn's burst damage at lower levels. You're rarely chasing any enemy in the early game for more than 3 seconds, so the total effect is generally is an added 160% of your attack damage.
Mejai's Soulstealer dramatically affects Evelynn's ravage damage, key to those one-hit ko and run situations (It also works into your mid-late game Lich Bane).

Tip: In 90% of situations(not all!), use ravage when exiting stealth in order to proc your Sheen/Lich Bane to massively increase your burst damage. NEVER use Deathfire Grasp to drop out of stealth with Eve, because it does not apply on-hit effects, but will leave you naked as Adam was when he met you in the Garden.

These four items are your best bet for nuking and escaping late game as AP Eve.
http://www.mobafire.com/league-of-legends/item-purchase-planner#&item=74-104-115-114

Lich Bane and Rabadon make you nuke harder.
Hextech allows you to backdoor and continue fighting even when you can't seem to get any kills.
Zhonya's Hourglass allows you to stealth, go invulnerable, and RUNNNN!

Feel Free to purchase other situational items such as Thornmail, Banshee's Veil and Quicksilver Sash.

Eve is great for killing stragglers. Some of them, it will serve your team a lot better if you leave for team fights. Eve is awful at killing enemies who can survive her initial stun and heal quickly or outright block her stun.

Characters such as Dr. Mundo, Olaf, Blitzcrank and Pantheon are very difficult to kill without adequate knowledge of when their abilities have been used and extremely close stalking to know when those abilities have expired. As such, don't put them at the top of your priority list if you're not sure you can kill them.

Warwick is one of the characters you should be most wary of. If you cannot burst him from stealth, you are probably dead, because his ability to heal will often overpower your ability to deal damage if you've already used your ace card (Ravage). Ignite will provide you some ability to reduce his healing effects, while dealing further damage.

Knowing how to apply this MIT(Most Important Tip) comes from plenty of practice.

XXXXXXX-Don't start a fight that will end badly for your team.XXXXXXXXXXX

Period.
If you see a half hp Mundo in the middle of teamfight, and his health seems to be regenerating. DO NOT SOLO HIM.
If you see an innocent little Twisted Fate with Oracle, DO NOT engage until he uses his next Gold card.

The best Evelynn player only dies for a good reason (e.g. sacrificing to drop the last nexus turret, stunning enemies to save players that have more situational value, turret suiciding to avoid giving a kill in the enemy base) Even fully committing in a team fight for Evelynn (may be) silly, as her stun-burst nuke is (almost) the extent of her team fight ability given strong team compositions.

Clarity and Ignite are what I use, clarity so I can continue stealthing in the early game.
Suitable alternatives include Cleanse, Ghost, Teleport, Flash, Exhaust, Clairvoyance.

Tip #198: Adapt your playstyle to current counters to Evelynn. If the trend is for people to ward the river bushes, get an oracle early or use Clairvoyance to find enemy wards. If people are getting oracle early, level to 6 fast so you can catch them with your ultimate. If people are buying banshee's veil, use your teammates to drop their spell block.

This basic guide to Evelynn was funded by spare time and a bad work ethic.
~Barnabyss~

P.S.

If anyone knows how to properly code the in-text items, please let me know.