You basicly have to decide what your team needs.
You decide between:
Tank. Your team needs someone who can soak up dmg, and your team has enough dmg/has a way to kill the enemy AD without problems. You build something like Wriggles or Dorans Blades -> Aegis -> Warmogs -> situational items like Randuins, Maw of Malmortius, Guardian Angel or Triforce/Bloodthirster if you need more damage.
Assassin. Your team needs someone who can assasinate and kill one enemy carry (AD or AP). Probably the case if your team has several tanks, but noone who can get to the enemy AD with ease. You go something like Triforce -> Rageblade/Gunblade -> Rabadons -> Deathfire Grasp-> Guardian Angel -> Abysal Scepter etc. Any AP item will do really, you can either go straight for AP, or you go for hybrid items (wouldn't recommend that).
You have to decide what your team needs, but you also have to consider what you like playing. If you like to play an assassin type of champ, build Eve the second way, if you like to be a tank, go build Eve tanky.
About the 2v2 lane: even tho its not really usual in the current meta, if you have to play a supportive role in lane, get 2 or 3 GP10 Items (if you have success in lane 3, if you are far behind 3, if its equal 2), I'd recommend Philospher Stone first since you can use the mana reg. After that either build Kage's Lucky Pick for more dmg or Heart of Gold for more tankiness. Consider buying Boots of Mobility, which will give you the speed to roam around and gank all other lanes .
AP Eve's goal is to kill one carry in a teamfight any maybe survive. Assassins are pretty bad right now (the meta simply isn't built for assassins such as Akali, Talon, Fizz etc - they see very little to no play), that's why Eve was really bad prepatch (yes she was UP as well!) and that's why people started using her as tank.
She has a somewhat reasonable amount of CC (only slows, but still), a very good steroid (good gapclosing!) and, due to her invisibility, she can initiate really, really fast without giving the enemy a chance to react. If you have Shurelias + W(active) + mobility boots, her initiation is strong because you give the enemy pretty much 0 chance to reposition or to disengage (except with a really good disengage like Janna or Gragas Ult).
Tank Eve isn't a good decision really. She has some strengths, but you got champs out there that do her job way better. But since assassins get countered way to easy in a "normal" Elo, tank Eve is the only way to play Eve right now if you want to win.
The problem with assassin Eve is that you can't 100-0 someone before they can react. If you run up to an AD carry and try to burst him/her, the whole enemy team will just turn and burst you down immediately. Tank Eve is only viable because every champ who can initate and can build aura items be somewhat successful.
You can still play Assasin Eve with a pure Dmg (AP) build, and its really a lot of fun, but you will,
Notice a major decrease in your win ratio as soon as you hit ~1400-1500 Elo
Always have the chance of losing, even if you're ahead by a lot, simply because assassins snowball, but they get punish severely for every mistake.
EDIT2 - The differences between AP and AD:
Building AP means you have to burst someone, cause as soon as you try to put out DPS with your Q, the AD/AP carries can turn on you and kill you. If you notice the enemy picking either high mobility or really tanky champs, dont play Eve as an AP burst-caster. Your job is to kill squishies, and if everyone can run away or is to tanky, you have no target.
Tanky AD Eve scales better since she is always able to tank. An AP Eve will fall off, because AD carries get so powerful late game. AD/tank Eve actually can and will work if played correctly (someone streamed only Eve and reached 1900 elo PREPATCH!). You have a huge impact early game (good ganks) and with an early Triforce (or at least Sheen), your dmg will stay high all early-midgame.
If you really like Eve as a champion and you want to play her, I'd recommend playing her AD with a Triforce -> tanky items -> Bloodthirster for more lategame damage, or Last Whisper against heavy armor teams. But as I already said, there are other champs that can do Eves job better (Rammus, Maokai, Nautilus).
The diffrences in the playstyles are pretty huge, and thats something you should always keep in mind. Always think about what your job is/will be: to tank and just CC the enemy down or to burst down one enemy and gtfo?