Hi, I am Thinker. I have been a Kayle main since June of 2014, i used her to carry myself bronze 3 and finished off in Platinum 5 by the end of Season 4. For context I started playing league in December of 2013 and ranked in march of 2014.

Here is a montage of some of my favorite kayle moments on stream from season 4 Play Video
I finished off season 4 with over 350 games on kayle, over this period i have learnt a couple things about this champion and this guide is a way for me to share this knowledge

Kayle has a very unique kit which allows her to fit seemlessly into almost any role in the game but with the current meta and the meta for the past couple months, the jungle is her strongest position.
I started of playing kayle mid but with the severe nerfs to her AS, it made a early game pretty week which meant her mid game would also be quick because you fell behind early. With the onset of bursty assasins, laning phase was painful pre-6. So i moved top and soon the meta shifted to kayle top, but in solo q kayle top has a glaring weakness; Kayle has to shove the lane and with no dash or blink, you become a prime candidate for their jungler.
This early game problem became a real bother until i started experimenting kayle jungle, my reasoning being : i can hide in the jungle until i am past my weak early game. Or so i thought, with the new jungle and items Kayle no longer has a weak early game. With proper proper damage management and route which ill get in more detail later on, you can easily clear, go buy, hit 4 and duel almost any jungler.
You also don't have to just farm, with chill smite and Reckoning with Divine Blessing to gap close or heal your ally, your ganks are very lethal. Not to mention with heavy handed, red buff and Divine Blessing for sustain you can solo dragon at level 5 if you've completed your Stalker's Blade - Devourer.

In most scenarios you want to use this rune page, as it helps your sustain in the jungle and with most jungles being AD, it helps with early game dueling.
On the other hand, if the enemy team has more than 1 source of heavy or have true damage champion like darius or irelia, this rune page would serve you better mid-game to late game. Keep in mind, that this will make you much more squishier early, so avoid unnecessary fights.

This part is pretty straight forward. As kayle jungle your role is to keep your teammates alive whilst dishing out ungodly amounts of damage and ensuring your teammates do more damage. You are basically both a carry and a support simultaneously.
Righteous Fury and Divine Blessing are your best friends for this task, you obviously want to max Righteous Fury for that next level wave clear and if you are going the on-hit route, it's your only source of damage. Quickly followed by Divine Blessing for that supreme sustain for you and your teammates and also to gap close. Reckoning is only useful for its slow, coupled with chill smite and Divine Blessing you will stick on targets like Irelia on Ashe. As you wont be building a ton of AP or AD, your Reckoning will be doing negligible damage and with it's immense mana cost there are no real reasons to invest a lot of points in it.

Even though you might be in the jungle, it doesn't want change the fact that you are still kayle. You are a hypercarry farming beast, never forget that. The aoe nature of Righteous Fury means you only need to attack 1 monster in order to clear the entire camp. Tax! Tax! Tax! i can't say this enough, if you go to a lane and don't get a kill ensure to tax your laner for your time. They might whine, scratch that, THEY WILL WHINE! but ignore them cause they'll thank you later when you carry their dumbass or save them with Intervention or Divine Blessing.
In mid to late game, keep your eyes peeled on free side lane farm and if possible a free turret. With your immense attack speed and sufficient AD [more so if you have Blade of the Ruined King you will melt through towers. The global gold can be seen as payment to your teammates for all their farm you took lol. You want to aim for finishing each game with about 200 cs and 2 turrets that you destroyed.

Due to my poor photoshop skills and even more laziness, i don't have like elaborate photos to show you. Instead i'll show you an excerpt from my stream with first minutes of a Kayle jungle game, which should give you a general picture of what path i recommend Play Video