Right. Before the flame starts, let me justify my choice. This is going to be a short guide, and it's just to change people's minds about eve. Here's the math:

AP Evelynn with (1 sorcerer's boots and 5 deathcaps):
AP: 1327.5
Ravage damage: 1647
Spike damage: 456.7
Total damage for 5 seconds: 1647+6x456.7= 2740.2

Yep, that is some crippling damage, and that's the reason why people play eve. Usually, no one goes 5 deathcaps. The usual damage is hence usually around the 1.5k range.

AD Evelynn with (2 fully buffed bloodthirsters, IE, PD, Stark's, and Zerker's):
Hits per second (with ulti): 2.5
Damage per hit: 379.1
Damage per second: 947.75
Damage per second with crits (50% chance, 250% damage) modifier: 1658.5625
Damage for 5 seconds: 8292.8125

Assuming Cho'Gath has no armour and stacked 3 warmog's with 6 feasts, EVE OWNS HIM IN 5 SECONDS FLAT. That's 3 times the damage output of the AP build, and it's a bit more feasible too.

So, yes, Eve AD build IS viable.

The items needed are as listed above.

With her ulti, Stark's, Zerker's, and PD, you reach 2.5 attacks per second (the max).

The 2 bloodthirsters, fully pimped, give +200 damage, while the IE gives an additional 75 with 250% damage crits. These guys add the bulk to her insane damage.

I can think of a few reasons why this build can fail badly. Firstly, if you don't farm well, you can't get the build, and you're dead. Secondly, most games don't last enough for you to get your max build. Thirdly, Eve is an early gain dominatrix. This late eve might mess up team dynamics and you may never blossom in time.

NEVERTHELESS, eve's invis, nuke, and ulti can net her quite a few kills early-mid game despite her lack of DPS then. If she reaches late game, she will be unstoppable 1v1 or 2v1s. E.g. Yi + Garen. Eve stuns yi, turns on ulti, and kills him in 3 seconds, and turns on Garen with full hp. He spins and silences, while Eve whacks Garen. Her insane lifesteal (60%, that's over 600 hp per second) will see her at mostly full health. Once the silence is over, she turns on ulti, and Garen dies in 10 seconds tops. Double kill in 13 seconds. In 5v5, she can just stealth in, kill 2 carries and walk out unscathed. Hence this build, once fully mastered, is probably the highest damage build of any LOL champion. I honestly do not believe that any hero, with any other build, can deal over 8k damage to 1 hero in 5 seconds.