I have loved Eve ever since I started playing her (which was during the time where she was considered absolutely horrible). Even with her so underpowered at the time I would carry a team more than one might expect. Needless to say I was overjoyed to see that she was getting a rework to make her more viable and ever since she has been updated, there have been few games with her where I did not get at least somewhat fed.

I will not be extremely in depth with my descriptions, but I will give you my thought process and let you know what has worked for me.

(P.S. requests or new ideas welcome)

Flat AD helps with jungling and basic damage - always a nice choice.
Flat armor helps with early jungle (as Eve is squishy on her own).
Magic resist per level as focusing on defensive items tends to make Eve a bit low on damage.
Movement speed Quints - great for getting around and ganking. (Other suggestions welcome)

Basic 21 offense for AD - Eve tends to be more AD focused now rather than the old possibility of AD AP or hybrid - and some defense to help with jungling.

Start with cloth armor and 5 health pots - basic jungle start.

If you do well enough (a gank or 2) you should be able to afford madreds razors and regular boots with your first back. If not, go for boots first.

Finish off wriggles.

Turn boots of speed into boots of mobility (great for ganking around the field)

Next is Black Cleaver. Start with the dagger as you usually will not have enough money for BF sword this early, but if you do extremely well you might.

sidenote: I have not seen another Eve yet that builds black cleaver here, I usually see trinity force. While trinity force is a nice item, and the next in the build, it simple does not give enough AD for its cost early game, and having more AD for sheen is always good.

Build sheen -> phage -> trinity force as sheen creates massive amounts of damage for Eve and phage is great to stop enemies from running and adding a bit of survivability.

At this point, you have decent damage but are still squishy, so build Guarding Angel. Start with either armor or magic resist, depending on where most of their damage comes from. (Banshees veil and thornmail also good choices, but only if their team is extremely heavy on one type of damage)

Next build up Bloodrazor. It may not seem like the greatest item on Eve, but with the attack speed bonus on her Ravage, this makes u shred through health at the start of a fight.

Finally, sell your wriggles for a bloodthirster, jungle should be easy now and more AD and lifesteal is always nice.

Hate spike is your most important and spammed skill (plus with the remake it does good damage) so max it first.

Ravage is your other main damage ability, with good initial hit and an attack speed buff this should be maxed second.

Dark frenzy is your mobility skill, and can work as a mini-cleanse for slows. Take one point early (I prefer at 4, when I finish my initial jungle route) and max it last.

Agony's Embrace - the amazing replacement for Eve's old ultimate. Obviously grab at 6 11 and 16.

Smite is a must if you are jungling, Eve needs it.

Flash is always a good choice, very nice for flashing into a target from the edge of you stealth range if you feel he would get away otherwise, or just a good escape mechanism.

Exhaust is also a good choice, very helpful for duels and gives a good slow for early game ganks.

Basic jungle route

Wolves

Blue

Wraiths

Double Golems

Red

Wraiths

Wolves or gank

Also, earlier ganks can work, but only if their team is very pushed in a lane, allowing you to stealth behind them or into their own bush (as you should attempt with every gank)

Early in the game you will look to get into lane behind an enemy, keep them at the edge of your stealth, and as soon as your allies move in, you strike. Unexpected enemies from behind not only hurt, but also avoid attack directed forward and usually cause hesitation in an enemy. Warning: if their team has some sort of trap mechanic in lane (caitlyn, teemo, nidalee, etc) do NOT go through the bush into lane. Remember: normal wards cannot see you, and neither can your enemies until you are on them.

After laning phase ends try to pick off any enemy that is alone, especially carries. With the AD from black cleaver and the sheen effect, you should be able to get a carry halfway almost instantly, allowing you to finish them easily.

For actual teamfights, try to be behind their team in lane if they have nothing to detect or hit you behind them. This way, an initiation from either team allows them to focus on your team while you sneak in and eat a carry or two. If being behind them is not an option, the jungle works as well, just be wary of traps and enemies entering the jungle.

When entering a teamfight (hopefully from behind) attemp to use Agony's Embrace on as much of their team as possible. This is ability is amazing, as it does 25+% of their entire team's health and then gives you a nice shield based on the amount of champions hit. You will then want to focus their biggest damage carry, usually an AD ranged, but sometimes the APs are more dangerous. Ravage for initial damage and attack speed, slap that Q, and if dark frenzy is up (since you probably used it to get into the fight) save it if you need to chase or run.