Hi everyone!
This is a short quick build I want to throw out there for those who are somewhat confused about building the new spider champion Elise! She is a powerful champion and rather unique in the game with a combination of skills quite unlike any other. Yes, there are other transformation champs like Nidalee, Jayce and Shyvana but none of them play anything like her. If you have no trouble playing her, great! Try my build if you like, maybe it would suit you better. For those who are finding her difficult at the moment, have a shot at using this guide and hopefully you'll find Elise much more rewarding.

Keep in mind though that I am putting this out there as fast as I can, I'm meant to be studying for exams at the moment but I couldn't resist sharing this build. Try not to down vote me too much, but I will be adding to the guide over time. Hope you enjoy it! :)

I build [Elise] as an Assassin with enough defenses to survive. You have to keep in mind in team fights you play her as a hit-and-run sort of champion (SLIPPERY LIKE A SPIDER) so you don't get melted. You 1v1 without much problem, but you don't have enough burst to completely decimate someone in 1 second either. Therefore, wait for the right time to pounce on a champion with Rappel and let your spiderlings feast on the crumbling enemy team, you will not be disappointed.

Laning, Elise is decent but does not shine in that much. She has a little sustain, she has escape, she has enough range to harrass, but she isn't someone immovable like Yorick or someone infinitely annoying like Teemo. Elise should always be in a solo lane, either as Mid or Top. I find top lane to be more safe as she has an easier time kiting melee champs up there, rather than being outlaned by superior mages in Mid (she isn't bad however). Jungling Elise is quite nice too, but I have not been experimenting with that so much.

After experimenting with both AD and AP builds and finally Hybrid, I believe that Hybrid Elise is by far the most efficient way to go. This is only my personal opinion. Here is why:

SKILL STAT SCALING

Building ABILITY POWER
On paper, Elise's spell scaling doesn't look too shabby. A strong 0.8 AP for her W in human form, mixed with scalings across the board. However, the other scalings give magic damage as percentage of health. For example, 0.02 AP in her Q is quite pathetic. 100 AP gives you an extra 2% of health in a supposed 'nuke', 500 AP (a decent amount mid-late game) gives you 10% extra. From 10%-18% is pretty shabby for 400 extra AP if you ask me, especially since the enemy's magic resist would counter most of your damage gains. By the time you achieve this much AP, your only reliably strong skill is your W, which can often be easily blocked by minions or dodged out of range. Your spiderlings do scale okay with AP, but keep in mind by neglecting your other stats it is not worth it. Don't get me wrong, AP Elise isn't as bad as a [Syndra], but it isn't the best.

Building ATTACK DAMAGE
Attack damage/Attack Speed is great in your spider form. You can take out large chunks of health and it is beneficial for W healing. You have great mobility with your Rappel, your stun is great for utility, and your Q and other percentage based spells aren't useless (as they still deal a base % of health). However you lose some needed damage on your spiderlings, you neglect your strong W, you have less heal, and you just aren't efficient. Yes, although AD can work, you waste a lot of stats and scalings. There are simply better AD bruisers/carries out there.

Building HYBRID (MIX OF ATTACK DAMAGE + ABILITY POWER)
This is what you should be going for. A supple amount of AP that gives a good boost to all your spells and your delicious spiderlings, it is beautiful. Moreover, you will absolutely destroy in your spider form with huge attack speed, huge sustain from both sides of the spectrum and solid attack damage. Another pro is the fact that hybrid builds are harder to counter through building defense, the enemy would have to stack defenses on both sides to see noticeable differences.

You are going to have to play careful early game (keep in mind you aren't weak damage wise, but you will go down quick). In any 1v1 situation you aren't likely to lose out so long as you can hit back.

I take 30.0.0 pure attack masteries (yes this seems a bit ridiculous) but I find this necessary for enough power on both sides of the spectrum. An early Elise can farm, kite and harass in human form rather well, so as long as you play safe before level 11~ish (where the bonus defenses in spider form are noticeable), these full offense masteries are well worth the wait.

I'll be giving a quick run through here (As said before, will add to the guide in the near future)

Start with Boots and 3x Health Potion. The mobility and sustain make up for your squishyness and is enough to last you in a solo lane until level 6.

Once you can, rush that Sheen. Suddenly, your damage will see a big boost through your basic attacks proc'ing the Sheen passive (eg. Q in Spider Form). Not only this, but it gives you mana and a punch to all your spells.

Next, depending on the opponent, finish your [boots of speed] into beserker greaves or Mercury's Treads. If you are doing well, finishing Trinity Force before tier 2 boots may allow you to snowball faster.

Buy a Vampiric Scepter and then finish a Hextech Gunblade. By this time, you will notice yourself healing like crazy when you W in spider form, and it offers you great sustain in other times too. It also adds good damage to both sides of the spectrum.

Now it's time to build defenses. Guardian Angel is just an amazing item in not only its passive but both its stats in armor and magic resist. It should offer you solid survivability, especially in spider form, you should be able to tank enough damage before Rappeling away!

The finishing items are up to you. Black Cleaver melts a lot of armor in the enemy as well as giving you good AD damage and attack speed. Rylai's Crystal Scepter gives you a good chunk of health as well as solid Ability Power and a slow that ensures you your enemy won't be getting away (if the Trinity Force slow isn't enough!). Abyssal Mask, Madred's Bloodrazor, Maw of Malmortius are all solid choices as they provide you that extra bit of defense while adding a sizeable amount of damage.