I have 2 builds on here:
My traditional AD/burst/Assassin Evelynn, (This is the one with the most use)

My newer, AD/hybrid/carry/Jungle build (this build has much less burst than the other one and is not meant to be as much assassin, although it can still function as such.) This one excels at pushing lanes and such as well as traditional team fighting.

First, before you go jumping the gun, trust me. this build works, I use it all the time with eve, and I greatly prefer it to AP burst eve. Now, for those of you who don't just trust honest opinion (I know you're there because I am one of you) here are some stats. I am taking the AP eve stats from the top rated AP eve build: Evelynn - An explanation by Calfurion (His build is great, I just prefer mine) <http://www.mobafire.com/league-of-legends/build/evelynn-an-explanation-22131>
He has: 728 AP (endgame, full mejais, including deathcap's passive)
I have: 80 AP
He has: 107 AD
I have: 424 AD (no stack items)

Initial burst (3 hits) dmg and secondary effects (just under 2 seconds, if using her ult):

AP Eve: 1048 magic dmg (Ignoring 32 MR) + 835 physical dmg (ignoring 26 armor) + 107 physical dmg (ignoring 26 armor) + 107 physical dmg (ignoring 26 armor)
AD Eve: 400 magic dmg (ignoring 0 MR) + 1060 physical dmg (ignoring 41 armor) + 424 physical dmg (ignoring 56 armor) + 424 physical dmg (ignoring 71 armor) + 300 magic dmg (

Hextech Gunblade activateable ignoring 26 MR)

so, discounting armor and magic resist, we have:
AP burst dealing 2097 in the first 2 seconds
AD burst dealing 2608 in the first 2 seconds
AP sustained dealing about 290(hate spike -26 MR/32 mpen) + 107(Auto attack -26 armor) ~(400 dmg/sec)
AD sustained dealing about 107(hate spikes-26 MR)+ 424(auto attack -71 armor) ~(531 dmg/sec)
Lich Bane bonus dmg (835 with -26 armor)
Trinity Force bonus (1060 with -71 armor)


Now, keep in mind with eve's most recent nerf, her Shadow Walk no longer stuns. The stun is vital for AP eve because she has little or no survivability. However, with my AD Eve's 40% lifesteal and 25% spell vamp, having a 3 second slow is just fine.

So, facts and figures out in the open, you tell me which has better burst.

The first Item I build for is

Tear of the Goddess. This gives eve a near infinite mana pool from early to late game, meaning you never need to worry about using spells once you grab this.

Second, I go for Mobility Boots, these give nice movement speed and allow for lots of getaways or chase-downs.

Third, I go for Hextech Gunblade. The active, along with nice dmg gives a 50% slow at a larger range than any of eve's other abilities. On top of that, it gives Ad, and some Ap (both help immensely with survivability) and with her dmg output.

By now, tear is about full (due to constant *free* lvl hate spikes every since it was purchased) and so upgrading to Manamune is cheap and still gives a good deal of dmg. (1100 gold for about 60-70 dmg)

Next is Trinity Force (starting with Sheen of course) but usually, I finish Black Cleaver before I finish trinity. (having sheen ASAP is important however)

Next, Black Cleaver. It quickly helps Eve shred her target's armor (for the team's benefit) and adds some nice dmg and attack speed as well.

Last item is pretty much up to you, depending on who you are fighting; my usuals are:
Bloodthirster: nice dmg and lifesteal. Great if you are not dying much and want the additional dmg.

Stark's Ferver: If the other team is builder lots of armor, this helps. It also gives your team regen, life-steal, and attack speed. With this, your initial burst leaves the target MISSING 91 armor. See how anyone but a rammus enjoys that!

Guardian Angel: If you find yourself the focus in fights and your lifesteal can't compensate enough, save your Shadow Walk and pop it right before you die. With GA active, your Shadow Walk will proc while you res and so you will return to life, already invisible. Great getaway and there goes the other team's opening combo.

Sword of the Occult: I have never been a fan of stack items, but whatever floats your boat. If you want it and can keep stacks, the damage and movement speed are awesome. Just make sure to grab it early if you are going to get it.

Infinity Edge: Adds nice dmg and great crits. Getting this as a last item can greatly increase your burst by adding a crit on the trinity force hit (with IE your trinity force hit will crit around 2.3k still going through 70+ armor)

Early-game is dangerous for eve, and you need to be careful not to get pulled in too far when you aren't able to take hits. I leave base grabbing my Shadow Walk, meki pendent, and 2 hp pots. Be careful in lane, especially if your lane-mate is melee. Watch for taunts and stuns that will kill you quickly early game. In essence, try to last hit creeps as much as possible, so that you can get tear of the goddess the first time you go back to base.

If possible, wait to go back until you have enough you finish your tear, and grab boots (~955 gold). Or, if you need the regen, grab Vampiric Scepter instead of Boots. Either way, tear is #1 priority so you can fill it up fast. Once you have boots, go straight for Bilgewater Cutlass. This gives cc/dmg that makes ganking possible and effective. Now grab Sheen for burst/sustained dmg. Once you have that, finish your Mobility Boots and move towards Hextech Gunblade. If possible, get the Hextech Revolver quickly and then proceed to gank whichever lanes need it. Keep in mind that you can tower dive if you will get the kill (because of your ult, despite the recent nerf).

Hextech Gunblade has about double the range and dmg of Bilgewater Cutlass for it's active, and makes for incredible ganking. It also gives you the survivability needed to get things done, and stay in lane afterwards.

You should finish Manamune whenever you have the time, because by this point it is giving you at least as much bonus dmg as a B. F. Sword, for just over 1/2 the price. The bonus will continue to increase until you fill it up the rest of the way.

After this, go for black cleaver. By now you should have decided what you want to build as your last item, so start for it, and/or finish Trinity Force.

Ghost and your Agony's Embrace make an awesome speed combo when you either need to get away from an ambush, or to catch someone who just barely got away. I like to save them both for when they are sure to make a difference between life and death (for me or my target), and use the accordingly. Teleport is great for map mobility, especially when well placed (on a ward behind an enemy or one of teemo's mushrooms for example). Keep in mind if you start your Shadow Walk before starting your Teleport, you will get there while invisible. (awesome for ganks from behind while you are at base or on the other side of the map) The same principle applies for a safe recall: Shadow Walk, then click the button before you disappear. (be fast at it)

Attack combo is fairly simple once you have cutlas/gunblade. Go in while invisible, (If you are sure youcan get a kill, or just about, pop your ult before you hit as it doesn't cancel your Shadow Walk) and get your first hit off with your E, dealing nice damage as well as lowering armor and MR. Once the fight starts, never stop clicking your Q as it doesn't interrupt attack or movement at all. If they look like they might get away, or you need the extra dmg, or both, hit them with the gunblade active. (I put mine in item slot 1 and hotkey it to G for ease of use).

Make sure you go for the squishiest/most dangerous enemy first in a fight so your 2600 dmg burst kills their AoE mage before he can combo. If possible, save your gunblade active for after the first part of the fight to save an ally,or yourself, or to ensure a kill on a running enemy.

INVISIBILITY NOTES:
Your Shadow Walk has a slight delay before you disappear, so get used to timing it correctly all the time.
Oracle doesn't kill you like it kills AP eve, because your burst will heal you enough to bring you back to full life from 1/2 in a matter of a few seconds. Do be careful however of unseen enemies and the like as you still aren't invincible. Make sure your team goes in with you if the other team is grouped or has an oracle. I find however, that if a caster like lux is hanging back (even with an oracle) save your Ghost so you can run in, kill her, and ghost/ult/invis out and get a kill and live 1v5 (I have done this many a time, against Lux. . . Annie. . . Ashe. . . it works. Just be smart)

BUILD 2 ONLY
Eve is not the strongest jungle, however it can be done; and her Incredibly high ganking potential once she has her Shadow Walk makes any difficulties in the first few minutes worth the trouble.
Needed:

dmg bonus from the offensive tree.
armor runes.
exp masteries.

Cloth Armor.
3 hp pots.


The best way that I find to start is to grab a Cloth Armor and 3 hp pots. This first sweep is risky and must be precise, but can help speed you up to grab razors from your first recall.

Have somebody give you a hand on blue (they will need either a heal or a hp pot) Smite immediately and as soon as the Golem hits about 400hp, the other guy needs to run immediately to lane. If you time it right, you can kill the Golem right as it turns aggro to you (you don't take a hit from it) with your E. You can then proceed to finish the small Lizards use an HP pot once you start taking dmg.

This will grow you to level 2 (grab hate spikes). Head to wolves and start on the big one. Once the first HP pot wears off, pop another one if needed. Head over to Wraiths and Smite the big one. Quickly dispatch the small ones using hate spikes. (Kill whichever one you want first, then attack the stronger one) Then head down to the 2 Golems and kill them (Use an HP pot when needed).

This next bit will get you enough gold for Madred's Razors, but takes risk. Watch out if anyone is mia. You will head up to the wolves in the other forest. Kill them using hate spikes lvl 2. Proceed to kill the two small lizards (if necessary, kill one, then get back to drop aggro then go back in).

Recall and grab Madred's Razors. If you have enough, go for 2 hp pots and a mana pot. Return to your wolves and kill them, then the wraiths. Grab small Golems, and if you have enough mana and hp, (1/2-full mana and 2/3+ hp) go for red buff. You can now proceed up to the other team's blue buff. At this point, you want to grab sheen and boots (and then Vampiric Scepter). With Shadow Walk at lvl 2 and these items, ganking should become very easy, and returning to the Jungle you can remain at full hp.

Waiting until level 6 to gank is slow, but allows you to get what is necessary to guarantee a kill; if there is an opportunity for a gank before level 6, the easiest way to take advantage is to pop Shadow Walk then port near the fight. Be mindful of not letting the other team see/hear the teleport.
With the great attack speed along with sheen/TF you can push towers incredibly fast. You have great sustained damage as well, but not as much burst as the AD burst build.

All in all, I greatly prefer AD Eve to AP Eve. Although AP is a very viable and effective build, I find that AD is more fun and rewarding. So give it a try, and when you carry your team to victory, ask them which is better. AP, or AD eve.