Passive: Spider Swarm
These are actually quite annoying for skillshot champions, often spiderlings will get caught in the line of fire for skills such as Ezreal's mystic shot or Kha'Zix's void spikes (my spiderlings sacrificed themselves all at once as I escaped with little hp). Remember that they can also tank objectives such as blue/red buff, dragon, baron and small camps as those enemies tend to target the closest unit regardless of if it is a champion or not. Making use of this effectively will greatly decrease how little damage Elise takes while killing stuff so she can use that HP in ganks. Spiderrlings also do a fair bit of damage early though their damage isn't as noticeable later in the game.
Q: Neurotoxin/Venomous Bite
MASSIVE damage for a spell, this Q combo is possibly more annoying than Lee Sin's Q combo. Unlike armor which most champions gain a massive amount per level as the game progresses, this spell does magic damage which is reduced by magic resist which is often gained at a slower pace or not at all for most champions. This means for most if not all the game, it'll probably be doing more damage than a Lee Sin Q combo.
In human form, casting this as your first spell (after Cocoon) is advised to make the most of it's ability to shred current % HP. It has a fairly decent range and unless the enemy has huge amounts of MR, will probably take a few hundred damage every few seconds on a target until they start getting below half. That said, this also destroys monthers despite having a bonus damage cap on it as you'll be casting this every 4-6 seconds anyway and the bonus damage cap is only for the % part so you'll still get the base damage in there. Also note that this drains quite a bit of mana in human form.
In spider form, this does huge damage to low HP targets so I advise on using it after you've done a Neurotoxin and a Volatile spiderling in human form to do maximum damage in your combo. Also this spell can be used as a gap closer as it jumps towards the enemy from 475 range. Using this to escape or dive on the enemy will make Elise much harder to deal with.
I often prefer to Cast one Q in human form during a gank, proceed with 2 bites and if they're not dead, switch to human form finally and Q again with a cocoon and switch back to spider if they're still alive somehow to bite them off. This is efficient as the spells have a separate cooldown and should be spammed often unless it'll let the enemy get away (hence why I like doing 2 bites instead of swapping forms after the first bite).
W: Volatile Spiderling/Skittering Frenzy
In human form, it's an additional damage spell for Elise, she can also use it to scout by casting it into a brush or around a corner though it travels fairly short distance before stopping unless it sees an enemy which it'll chase until it loses vision. The Aoe damage isn't bad for killing ranged minions or helping kill the smaller minions of a camp as you don't need much to kill those anyway. In combination with Neurotoxin, this leads the burst combo after a Cocoon before turning into the spider. Do not waste mana on this spell if you're getting low on mana and are ready to gank/fight soon as I tend to prioritize having mana for Cocoon or Neurotoxin as those spells are more useful than this one in a fight.
In spider form, this is a nice sustain spell to negate the little bit of damage you receive as spiderlings should be tanking. Fairly basic spell, early game it tends to do a huge amount of damage in duels as nobody is ready for the spiderling/elise spider form damage frenzy that occurs from this when champions have low armor/magic resist.
In human form this is a great long range stun which when/if it hits, can basically kill or severely hinder a champion due to them not being able to get out of the way of a combo as they're stunned for a bit. This will usually lead the combo as the opponent will be immobile for you to human form Q/W, R into Spider, Q/W and chase down with another Q then possibly a R/Q/E if cocoon is up again. Note that missing this is very bad for a combo as it's cooldown is fairly high and the enemy will be able to run or dodge the volitile spiderling as well as possibly getting out of range for neurotoxin.
In Spider form this spell is fairly basic if you've ever faced a Elise. Elise can jump to wraiths/minions/champions and such to evade capture if needed. Also useful as a skillshot evade to merely jump up in the sky then choose to land on the enemy Elise pleases to devour. Using this spell as a chase tool is fairly simple, always make sure your cursor is on the target you want to jump to or else Elise will simply jump in the air and the target will get out of range if they're running, but if you manage to click the target you want while they're in range, Elise will jump a huge distance towards them even if they keep running. In other cases this is useful to find a target that ran into a brush as you can jump on them if they're still in the AOE after you scout them when you're in the air.
R: Spider Form/Human Form
In human form you get range which isn't that useful for jungle Elise aside from placing a redbuff on a opponent before swapping forms but spider form gives spiderlings which are a fairly nice bonus as they can block skillshots and do quite a bit of damage if given time. Also note that I like traveling in spiderform around the map whenever I can because Elise gains 10 movespeed in spider form making it an ideal tool to walk around the map unless you're about to fight an opponent in which then you should lead with human form's Cocoon.
Fairly straightfoward, proper use of this to do combos is key for Elise players. Like I've stated above, my favorite combo style is:
Cocoon->Neurotoxin->Volatile Spiderling->Spider Form->Bite->Frenzy->Rappel if enemy out of range->Bite->Human Form->Neurotoxin->Cocoon->Volatile Spiderling->SpiderForm->Bite->Frenzy
The reason for doubling the bites instead of swapping forms for another Neurotoxin is due to the gap closing ability of spider form's bite which allows us to continue fighting with spiderlings and maintain the distance against a champion that is fleeing.