I have been experimenting with it. I kinda like it and it works but its not the jungler for me and it needs alot of practice time. I mostly have been playing Kayle ap mid. But in the jungle, I go for more of an atk speed build. Remember your E does bonus physcial damage equal to 40% of your ad.
I pick up items like Spirit of the Lizard elder jungler item, the Zeypher, Nashor tooth and atk spd boots. After that point it is kinda on what you need. Need more magic damage? Pick up a death cap or a haunting guise. Maybe you can build a warmogs or more ad. Its really maters on what your team needs because Kayle can come out of the jungle with any build.
It doesn't mater to much because Kayle can clear the jungle so fast that my smite isn't even up to take the blue or red buff.
As for runes and masteries, I can't really say which is the best. I would probably go 21/9/0. But do you go the ad route or the ap route? Do you go ad runes or ap runes? I can't say because it all works. I know Snooopeh from EG had armorpen on his reds and maybe his Quints and no ap at all. so that means he went down the ad route in the masteries. It worked for him. He maxed E first and made the most out of those rune choices. I've been running AS for reds but maybe I will switch it out because she clears so fast already she doesn't need it.
In the composition and team, Kayle is far more of a...Kayle, than a jungler.
You can jungle, and your clear, saftey, and ganking is all a solid middle ground, so you won't fail or excel at anything. Kayle however, can function as a champion, with little to no gold. And no mater your build you're able to scale well enough to be damn strong when Fed. But if you don't get fed, you've still got great utility which can be used.
When I first started playing league, i only saw kayle jungle. i dont know what the meta was at the time, but i didnt see kayle in a lane, except to support sometimes. Nowadays, AP kayle has been popularized by pro play, and many are sending her mid. I still think the old school hybrid kayle with the new season 3 changes to items is the strongest, but league is a game about variety and personal preference. If we didnt get to see interesting and completely non meta things like Saintvicious Sona or Ezreal jungle, league would just be much more boring. Kayle jungle still "works" and can be good with a proper team comp.
Just build lizard elder and zerk greaves. After that it is all situational. Zephyr is really good on her. You can go ap kayle with sorc boots and nashors but lizard elder just tears through camps with some attack speed plus true damage on champs is really good. Clearing can be a pain until you get your first item though.
I usually take her offensive hybrid jungle masteries. 21/9. shes got great dps, okish sustain, and great late game support and dmg.
i go for lizard if their team is suseptable to true dmg (like not a bunch of health tanks, in that case i get the claw part of lantern and leave it as is). lizard, zerk greaves, nashors, usually a liandrys ( it applies to every single auto attack.. and if you have yiur slow on does massive dps) frozen mallet , and BotRK. you have great % hybrid dps, tanky, CDR, and near everything kayle needs to function.
elder lizard> AS boots> Nashors> Liandrys> BotRK, FM.
I've been going Kayle jungle recently and having success with it. Kayle jungle as Misticaltom said isn't the most amazing ganker in the world, but she's not bad either. If your health is a little low from jungle, you can W to heal yourself as you enter the lane to both get your health up and for the speed boost. Activate your E (your ranged attacks) while closing in on the enemy, then Q them as soon as possible. Then just lay on your auto attack while the laner bursts them with everything they've got. You can also heal the laner (or yourself, if necessary) before going back to the jungle.
My rune setup for her is attack damage focused right now (AD marks and quints, armor seals, magic resist glyphs). I plan to get some AP runes soon, I just can't resist buying champs long enough to get more runes :P. But this setup works.
For masteries, I go 21/9/0. I do miss out on Runic Affinity (the one that makes your jungle buffs last longer), but I like having the extra health. No real surprises, but here's the breakdown (I'm just going to list what they do since I never remember the names and don't want people to have to look them up.
1/1 in Executioner (+5% damage to targets below 50% health)
1/1 in Summoner's Resolve (10g per Smite, use often. This is the reason I go 9 Defense instead of Utility. If you want to forgo this source of gold, go into Utility for Runic Affinity instead)
4/4 in Durability (+6 health/lvl)
2/2 in Tough Skin (-2 damage from monsters)
1/1 in Bladed Armor (6 damage to monsters who attack you)
1/1 in Veteran's Scars (+30 HP)
As for my item build, I usually forgo jungler items entirely. I'm sure it might make it a little easier, but Kayle can do without the jungle items as long as you get a good leash, and she can just start building towards her big-ticket items immediately. I start off the game with an Amplifying Tome and one potion. I usually play with friends so I can rely on them for a good leash.
Take your first skill point in E (ranged attacks + splash damage is invaluable). At lvl 2, put the point into W and heal yourself up as often as you need, with blue buff you can go ahead and spam it as much as you like (also helps in getting around the jungle faster). I max E first pretty much always. As for Q and W, it depends on how aggressive you and more importantly your team as a whole is being. If you're getting out there and ganking a lot, max Q first for that huge movespeed slow. If your team is being more defensive, max W to keep yourself sustained in the jungle and to get around quicker.
Anyway, I rush Nashor's Tooth as quickly as possible, easy since it builds out of the Amplifying Tome. After that I grab boots. I go either Beserker's Greaves or Sorcerer's Shoes, depending on enemy team comp. Recently I've been favoring the greaves for the extra attack speed, but if for some reason they're building magic resist (and let's face it, they're not), go for the Sorc shoes instead.
After boots I go for Liandry's Torment. The health and %health damage is amazing, and it gives a huge, noticeable damage boost once you complete it. Put your Q on an enemy to proc the full Liandry's damage. After that I got for Rylai's Crystal Scepter, just so that each auto-attack procs the full Liandry's damage.
After that, I finally (usually) grab a Rabadon's Deathcap to boost your now-substantial AP even higher. Your last item will depend on the situation of the game. The combination of Nashor's, Liandry's, and Rylai's gives you a ton of damage, so it's doubtful you need to build any more. I'd go for tankier items (Warmog's, Frozen Mallet, etc.) or spellvamp (Hextech Gunblade, Will of the Ancients, or if you did decide to go jungling items, finish up with Spirit of the Spectral Wraith (Elder Lizard isn't bad, but the Smite cooldown reduction and 20% spellvamp is awesome on Kayle, although if you went jungling items you should finish this up earlier when you're really using Smite a lot)).
So I know this turned into a huge thing, but I hope it gives some jungle Kayles some insight into how I play. Here's the cheat sheet:
Attack Damage marks and quints, Armor seals, MR glyphs
Typical AP in Offense, grabbing 3 points in Fury for the attack speed over Havoc, 2% increase in damage isn't worth it
In Defense, grab the Summmoner's Resolve for Smite gold and get the two jungling masteries.
I think McToomin has the build down pretty well. I can offer a few alternatives for items:
1. Guinsoo's Rageblade: This item is so gold efficient that, when all passives are active and you have full stacks, the item's gold worth is almost double the amount it cost you to buy it (more specifically, 4900 gold roughly in terms of gold per stats). It also synches with Kayle's E, and it can give Kayle sustain in teamfights. Its one weakness is that the items you build into it, a pickaxe and blasting wand, aren't the most efficient items to buy early on for jungling in Kayle's case, therefore Guinsoo's is a cost-efficient 4th or 5th item since you'll want Nashor's Tooth and Liandry's Torment before then.
2. Twin Shadows: A support item, it ended up being an excellent item for AP utility as well. It's cheap, built from a Kages, and has a useful active for recon purposes as well as slowing escaping opponents. Not the best item to buy if you end up getting fed, but it's certainly a major bargain for the stats.
3. Boots of Mobility: One of the best things since sliced toast with melting butter (and onion salt, yum) if you end up having to gank lanes alot. Even though Kayle has her W, it can eat up mana fast if you don't have blue to sustain the costs. These boots help with solving that little issue. You can always sell them later for the zerker's if you want to later on.
4. Sightstone: This item has mostly become a support item now, though a few junglers do make good use of this (looking at you Lee Sin and Jax on this one). A bit more expensive now, but it does give health and, more importantly, alot of wards that you will now have for the rest of the game. Some junglers can abuse bushes like there's no tomorrow, and if you have to duel them, this allows you to surprise them with the pre-emptive strike. Definitely worth its cost in gold, but not necessary to buy for a jungler that doesn't have a unique mechanic to abuse wards for escape/engage purposes.
5. Flask: The cost made it a bit more difficult to justify for some lanes early on, but junglers can pick this up for a nice form of sustain. It saves you on a few hundred gold worth of potions and overall it's more noticeable than a spirit stone sometimes.
6. Madreds: In theory, it is possible for Kayle to jungle without a machete, but if you rank solo q (or just solo q in general), it's a bit more dangerous to go without a machete since you don't know how well your team will help you to kick-start the jungle. Madreds is a solid item on Kayle early on, she's fragile and without her E, her ability to clear is one of the worst naturally. Madreds changes this, allowing you to potentially sustain your fast clears. You can also go a spirit stone if you want, but I wouldn't recommend any of the upgrades, they're a bit more expensive and they don't give Kayle necessarily the stats she fully needs for an explosive late game. Also, if you're one to favor Madreds on jungle Kayle, you might want to consider AS marks for more chances to proc the Madred's passive.
7. Locket of the Iron Solari: This is another support utility item, but there's a pretty good reason why alot of pro junglers are buying this item: the stats are so freaking cost efficient that some wonder if this is a worthy laning item for non-supports. CD, a generous amount of health, a nice shield who's effectiveness continues to scale with levels (though obviously it's weaker late game), for merely 2000 gold? Consider this item if you decide that you need to run a bit of utility early on and your team already has a good portion of damage, the locket's shield can save lives and deny enemy gold. Plus, the cd means more heals, more Q slows and more ult shields....
These are some items to keep in mind. Personally I favor gold efficiency alot, and since Kayle normally demands expensive items that can't all be bought easily through a jungle budget, sometimes you may need to look at a utility item or two to help save on costs early on. But, to be frank, once you have that Nashor's Tooth and Liandry's Torment, you're normally going to start hitting like a truck all the way to late game and beyond.
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EDIT, one thing i've also been doing recently is rushing Nashor's, starting with Stinger, and then Malady > Rageblade
Mid game build looking like, Spirit Stone > Boots(AS or MPEN) > Nashors > Malady > Rageblade
The spirit stone can honestly be upgraded to any of the 3 items, Spirit of the Elder Lizard is a good one, even though you're not building AD, the passive DOT is not that bad.
Although Liandrys is a GREAT item, I personally find the extra attack speed perfect for clearing camps, pushing lanes great. And having so much attack speed helps to stack rageblade pretty fast, rageblade being one of the most, if not THE most gold efficient item.
I'm nowhere near a Kayle expert, but I've been playing her pretty much almost exclusively this season, right now I have an almost 80% win ratio with 40 games.