I've been working on making Eve viable for my entire time on LoL. She has great pub-stomping abilities, but requires a specific situation to really shine. AP always had issues vs high-HP opponents, AD had issues with being focused down during teamfights (and slow to start). Dominion has changed that.

Dominion is much more about mobility and 1v1 (or 1v2) sub-battles. This is where Eve tends to shine the most. Frontal assaults are not her forte, but protecting (and stealing from) carries makes her a shining example of how to overwhelm the opponents.

Looking at her kit, she has the following abilities:
Hate Spike: Random(ish) DoT, as long as you keep mashing the buttons. AP-based, high rate of fire, but not very good AP ratios.
Shadow Walk: Her bread-and-butter. The "piece de resistance". The reason people rage, QQ, why Riot has repeatedly nerfed Evelynn until she is a pale shadow (haha) of her former self. Only has a slow (not a stun anymore, dang NerfBat TM), so could be good for escapes (or helping others escape), but has no DIRECT influence on either AP or AD. But read further....
Ravage: Ah, here's the good stuff. Does 320 (+100% AP) magic damage, and shreds Armor/Magic resistances by 26 for 5 seconds (at level 5). This is her nuke. Unfortunately, I do not believe the -MR helps Ravage itself, so the only thing it would help is either her Hate Spike, or her auto-attacks. We have already talked about the "random" nature of the Hate Spike, so that only leaves auto-attacks to fully take advantage of her own ability. We aren't going to talk about what it does for the entire team, since that's pretty simple to understand.
Malice and Spite: Every time Evelynn scores a kill or assist, she regains health. Activate for 75% AS, and 30% MS for 10 seconds. Refreshes every time an enemy champion dies. This is her chaser, her escape, the method by which she can pounce upon the lagging Veigar in the back and run away cackling while the opponent team tries to figure out to chase you or worry about the other 4 teammates who are in front of them. MS is always good, but the AS is wasted on a strictly AP build.

Now with Dominion, there are new on hit items that produce beautiful effects. Chief of these is Ionic Spark, as well as the remodeled Kitae's Bloodrazor. Both do magic damage, and if you add Malady and its own MR shredding ability, you can reach the max attack speed of 2.5 attacks per second when you use your ult. Top that with Shadow Walk's 3 seconds of slow (where you should easily keep up to the opponent) and Ravage's 5 second debuff, you have 5 seconds to whale away at an opponent and 5 seconds of speed to get away. IE: If you jump someone and can't defeat them in 5 seconds (or close to it), run off and leave them in bewilderment as to what just happened.

Malady is the core item to this build. However it isn't the FIRST item to build. Start with a Ruby Crystal, Boots, an Amplifying Tome, and three Health Potions. All three items are things you are going to upgrade early on, and are important to surviving (and winning) that crucial first tower assault on the Windmill. You will most likely upgrade to the Ionic Spark FIRST, then worry about the Sorcerer's Shoes to catch up to everyone else. Then move on to Malady which will give you a boost of damage right when things start slowing down for you. Finally, move on to the final attacking item.

At this point, it really depends upon what you are facing. Kitae's Bloodrazor is the default item, but there are good reasons for Wit's End (for MR) or Sword of the Divine (for the double "on four" proc and super-secret anti-Jax tech).

Finally, you have two slots open for "utility". Usually that means "Defense". Sunfire Cloak is damage (magic, of course) and armor. Thornmail is also magic damage, might be nice to see the interaction w/ the -MR you will be stacking, but you don't normally have the health to make it really worthwhile. Just don't forget it as an option. My normal preferred item is Frozen Heart vs high AS AD attackers. Slows them down, and about the same defense as the Thornmail.

Another good "utility" item is the brand new Priscilla's Blessing. Health Regen is never an ability to get excited over, but they stuck it with two other interesting abilities. Increased capture rate and increased move-speed on activation. A mini-Ghost, if you will. If you are having problems for some reason vs their team comp, or they are just defending turrets that your team is having problems breaking, switch gears and force the battle on your terms. Hit the lightly or undefended turret and capture (or at least make it neutral), then move on to the next location.

Malice and Spite is the primary ability to level (of course).
Second to that is Ravage, which is the second part of your damage engine.

Once you have enough Stealth to get into position (2-3 points), I would put all your points into Hate Spike until you have nothing else available to put points into.

This build works best vs squishy opponents. Even vs champs with CC (Jax, Xin Zhou, Veigar, etc), you should have little to no problem holding your own 1v1. However, champs with high amounts of health and that build almost all tanky items, you are better off finding a different target

Unless you build tanky defensive items, you are still squishy. Avoid initiating fights unless you find an opponent alone in a weakened state. You do not do well when outnumbered, so know when to withdraw when you are defending a tower by yourself. Force the battle on your terms: ie go capture a tower in order to force the opponents to separate, then drop on them while they are by themselves.

Do not worry overly about Thornmail or anyone with high amounts of armor. You may -look- like an AD carry, but you are actually a magic-damage heavy DPS carry. Thornmail and other armors will do little to you (with the exception of Frozen Heart, which can slow your attack speed). If you cannot kill opponents within the 5 second window, learn how to withdraw or pick better targets.

If you are in draft mode and see that the opponents have picked Morgana, Singed, Jarvin, Rammus, and Rumble, this is NOT the time to play Eve. However, if you see Ezreal, Kassadin, Xin Zhou, Tryndamere, any AP caster, or anyone that you would not expect to build tanky with heavy resists, that is going to be where you will shut them down.

Vs other high-burst assassins: Poppy, Akali and the sort, make sure to wait for them to try an initiation on your team or when they try to capture a point. Ravage them for the setup and then proceed to tear them apart. Make sure to have some sort of defense for their burst and just out-burst them at their own game.

I would strongly recommend getting both Ghost and Flash. Both can be used for offensive, defensive, and reaction. You will be able to chase down opponents, escape from ganks, and capture/counter captures. If you can't be there in time to capitalize on a push, a defense, a fight, you aren't doing your job.

Exhaust is an alternate, but it encourages you to stay and fight vs an opponent you are not matched well versus. If you are stuck whaling away at someone longer than 5 seconds, you initiated badly or picked the wrong target.