Introduction
Welcome to my Elise jungle guide! This is a fairly concise guide describing an AP jungle Elise build for the revamped Season Three jungle. I've played Elise frequently since she was released, but with all the Season 3 changes I found myself needing a completely new build for jungling with her. Season 3 has made Elise significantly more powerful in the jungle. The big change that improves Elise's sustain is a change with neutral monster aggro that allows Elise to easily divert monster damage to her spiderlings. Elise's clearing time has been improved by the new item Hunter's Machete and other S3 changes. As such, Elise is more viable than ever in the jungle. This AP build turns Elise into an effective jungler that retains the benefits of good AP damage. I have had good success playing this build in normals, going 3-1-21 in my most recent game. Read on for the Pros and Cons!
Pros / Cons
Pros
Fast clearing time
Impressive sustain with her spiderlings and
Volatile Spiderling / Skittering Frenzy
Good damage in team fights
Good chaser with
Cocoon / Rappel
Extremely fun champ
God-tier dance in spider form
Cons
Can be bursted down easily: she is not a tank
No all-around useful escape mechs
Requires a lot of skill to play effectively
Items
Hunter's Machete is an excellent starting item for junglers like Elise. It is astoundingly cheap and provides very useful bonus damage to monsters that you will use all game. Along with this pots are a neccesity for healing up while jungling or after ganks or fights.
Sorc's shoes are a must for many AP champs, and I consider it the best pair of boots for Elise. However, Berserker's and Mobility Boots are also viable, for increased damage in spider form and faster speed in the jungle. The Furor enchantment is very helpful for spider form. Q in both normal and spider form will trigger it, and the speed helps you track down enemies in spider form.
Rylai's is core on Elise. The health will help you significantly in fights and the slow is crucial to your core combo. The combo of this item's slow and
Liandry's Torment is killer on Elise especially.
The powerful burn on this item is nothing to scoff at, and extra ability power and health is always great, especially with some bonus magic pen
I'm still getting used to this item, but its a very good final item to build on Elise if the game progresses this far. It has everything Elise needs, AP, CD reduction, mana, and a bit of extra tankiness.
Ganking, Teamfights, and Combo
When ganking with Elise you have a variety of options. Two primary options immediately come to mind.
You can:

Start in human form, hurl your web for an impressive stun, quickly Q then W them, switch to spider and continue attacking, using Rappel when useful or needed

Start in spider form, rappel over, Q then W before switching to human form and stunning with web before rapidly Q'ing then E'ing
Pre-teamfight you can harass effectively with your human form W. When in teamfights, play it safe early on with your ranged human form abilities, then switch to spider to finish off stragglers or deal more damage when the time is right.
Your core combo is as follows, and is similar to the gank technique described above: start in human form with your E, then rapidly follow it with Q and W. Switch to spider form and rappel, following it with Q and W. If they're not dead, repeat. Keep practicing this combo and perfect it, as it is extremely important.
Conclusion
Thanks for reading my guide! I put a lot of work into it, so if you enjoyed or had success with my guide a positive vote would be greatly appreciated! If you disliked my guide, please provide a reason before you downvote and I'll do my best to address it. I'll also likely be editing this guide in the future to add other possible build items and more complex gameplay tips, as well as pictures and other pieces of flair to hopefully negate the several looming walls of text. Again, thanks for reading! :)
Elise Pro Tips
Following is a stream of random tips and ability uses that I've discovered while playing Elise. I'll likely add more of these in the future
With Rappel you can pull of insane escapes from fights. Remember Rappel allows you to move almost instantly to champions, neutral monsters, and enemy minions. In fact, Rappel is what gives this guide the name "The Agile Arachnid", as I feel it is something very unique to Elise. Rappel is also sort of like a free
Zhonya's Hourglass in combat, so use it accordingly.
Your Cocoon is a very powerful stun. Use it to shut down the enemy carry or snag a fleeing enemy, or anyone for that matter. It also provides vision of enemies hit, so fire it off into a bush you suspect someone is hiding in for a free initiate that can often lead to a kill.