Kayle is a much derided & misunderstood champion. She is among the most versatile champs in the game, capable of solo top, support and carry. Less well known is that she is a great jungler.

Jungle Kayle needs the following to function;
Attack Speed
Armour
Mana Regen
Cooldown Reduction

How you get these is up to you. There is no wrong way but you will eventually need a little of all of them (and a lot of AS).
Personally:
Attack Speed - Items & Quintessences
Armour - Seals & Masteries
Mana Regen - Items
Cooldown Reduction - Glyphs & Items.

An alternative starting build is a full AS run page (+38%) and Cloth Armor & 5x Health Potion. I don't choose this because I don't plan on building cloth armour into anything in my build. It gives 18 armour, 21/9/0 and flat armour seals gives 18.69. You lose 8% AS initially (buying dagger first) but you have armour penetration & cooldown reduction.

Pros

Fast Jungle clear
Good ganks
High damage early in game
Special/heal+speed means she is also roaming support
Versatile item builds to fit with team
Hard to counter-jungle after lvl 6
Leash not really needed on blue
Becomes carry-esque damage

Cons

Vulnerable to counter-jungling before level 6
Mana-heavy, needs blue longer than some other junglers
Slows but no root/stun
Non-tank jungler doesn't fit 'perfectly' with 'current meta'

Jungle route is variable, I tend to start on Wolves but can do blue first instead.

Wolves > Blue > Wraiths > Golems > Red

EDIT:
While the above works reasonably well I've found that the jungle run is smoother if you do blue first so you're level 2 at Wolves.

Obviously Take Intervention when available, otherwise...

Max Righteous Fury first, this is a free BF sword + more on your damage output. This is why you build AS before any AD, you already have lots of AD.

I have found that in games where I max Righteous Fury first and build AS with a B. F. Sword added later (maybe into Black Cleaver if the game allows), my damage output for the whole game is roughly 50/50 Magic/Physical which makes Kayle's damage hard to itemise against.

Reckoning or Divine Blessing?
The build here lists Reckoning as 2nd to max (though not to take until lvl 3) and W last. This order can be swapped.

Reckoning is great for ganking but the slow % doesn't increase. Divine Blessing's speed buff is great and it's heal component, while poor at rank 1 is considerably more useful at higher levels. I often level Q and E together. This is more of a personal preference thing.

I will suggest though that if you find your gank attempts narrowly failing often you might want to rank Q ahead of E.

Smite is obvious, use to finish a mob, not just as extra damage.

Second spell is much more variable. I like to go with Flash for the escape/wall jump but Surge is very powerful for Kayle also because both AS & AP benefit her.

Ghost, Exhaust & Ignite all work well too. Your preference really.

Several mastery builds are discussed with Kayle jungle.

The most common & my comments:
30/0/0
While it is true that Kayle is a hybrid champion, changes to her mean that AP does not translate to damage well. Not bad but not my preference.
21/0/9
I have tried this and liked it, just not as much as my preferred, partly because my item choices cover the mana regen aspects of this anyway so it is partially redundant. This would probably be preferable if you chose not to build Nashor's Tooth though you would still want the CDR from somewhere, perhaps The Brutalizer or keeping blue buff up. Early jungle also felt a lot more risky.

I prefer:
21/9/0
Early jungle is consistent thanks to the armour & faster due to the returned damage. Damage is high from all the necessary physical damage buffs from offensive tree.

Core item build
Berserker's Greaves
AS & movement speed. Nuff said.
Nashor's Tooth
The only AP item you'll be buying but you're buying it for the AS, the mana regen & most importantly the CDR. The AP is just a bonus!

Everything after these 2 items is somewhat dependent on how you want to build towards the end game.

If you feel you need some Tenacity I'd recommend getting Cloak & Dagger over Mercury's Treads because of the early AS & MS combo which is vital to fast jungling.

Tanky DPS
Frozen Mallet/ Warmog's Armor + Atma's Impaler

Carry-esque damage
Phantom Dancer + Black Cleaver

As always, use item builds as a guide only. I haven't included defensive items because their usefulness depends largely on your opponents.

Ganking
Kayle's ganks are simple but effective. Make sure your team know your target & are in a position where they can get involved. Charge out of the brush. Self-casting W is a good idea if you have a lot of ground to cover. If not save it for any chase-down that might happen. Activate E on your way into range & hit them with Q from maximum range. If possible put yourself between them and their exit route. The damage you can deal out in the next few seconds will often be enough to secure a kill. People are often confused by Kayle's appearance and forget to consider her like a ranged character and will run when they should fight.

Once the team fights start happening there's a few things you need to remember as Kayle.

1. You are SQUISHY. Kayle has low HP and not a massive amount of Armour or MR inherently. If you get silenced/stunned then you can't use your special and if the enemy is smart they'll focus you early if not first. To avoid this, never initiate or face-check a bush.
2. Your heal is best used for the speed buff or for saving people with DoTs on low health. It is rarely useful to be used as a normal heal.
3. You have one of the most powerful specials in the game. It will let you or others turret dive and it has a long range. Make use of these 2 things together by placing it on someone from a hidden place in the jungle as they turret dive where they normally couldn't.
4. Don't use your special on tanks, save it for your damage dealers.

If you ever find yourself in a team with Morgana & Katarina you can have a lot of fun in team fights. Morgana's Shield + your Special + Kat's Special = Invulnerable whirling spinning ninja of doom :)