Hey, sorry about the goddamn novel I'm writing here lol. Hope you make it thru. I also recommend you write down what you think are the important parts, because this wont be the easiest to remember. Or just copy paste this into notes or something. Lol lets get started.

Getting form, and Clearing jg camps.

So, 99% of the time you can choose what form to go before the game even starts. Your clear will change based on what form you want. If you want red form, clear to hit level 3 and be on the same side as a melee lane. The opposite is true for blue form of course. So I recommend starting red(yes kayn CAN start raptors but that doesn't always mean he SHOULD. Sometimes its better to get a leash for a healthier clear to level 3 gank. After red, if you want to gank the lane on your red side, take raptors, then krugs(Q 2 times then smite the big krug, then keep Q-ing) then gank. If you want to gank the lane opposite to your red side, take red, raptors, wolves, then blue. Use your E to sustain you when you can. From here you can gank, or take gromp and scuttle. My personal clear is the same if I'm early ganking my redside laner but for blue side its different. Kayn can full clear and get to scuttle at 3:15 exactly. Red, krugs, raptors, wolves, blue and gromp. It's a very hard clear to do properly, and be fast enough, so I just recommended the basic clears however if you wanna know more about this one then just ask.

Ganking

NEVER FORCE A GANK. If it won't work, it won't work. Take the ganks you know you can. And remember your W is a 90% slow so if you land it early, that will let your ally whom your ganking for catch up for the 2v1.

Ability maxing and usage

Always start Q(unless your doing that cheese where you shadow step into the enemy's blue and smite steal it, but that's a low elo cheese strat. Start Q mostly.) Then 99 times out of 10(see what I did there? Lol) you will take E second. Its mobility yes, but most important, its good sustain in the jungle. W 3rd. When going red form, max Q then W then E, leveling ult when you can, at 6, 11, and 16. Your Q when maxed is on such a short cool down, it's up basically every 2 autoattacks(if your good you can sometimes cancel the animation of the second auto attack with your Q animation. Hard to do and not needed, but every little bit helps.) and as Rhaast(red form), a huge chunk of your healing comes from your Q, due to it hitting 2 times, so you want it on as low a CD as possible as fast as you can. Shadow Assassin(blue from) however, after you get all 3 abilities, you put 2 more points into Q, for a total of 3, then max W. W gains more damage per level then Q and you dont need the extra ability haste on your Q because you will likely only use it once or twice in a fight then run away for at LEAST 8 seconds due to blue forms passive. You can max Q first then W, it won't make much difference. Plus i haven't done the math in a few months it may be wrong now. Its purely personal preference. Side note. Because around 30-40% of all Blue forms damage is dealt AGAIN as magic damage for 3 seconds, it adds up to a pretty decent chunk of magic damage. If you don't feel like you need mobi boots, you can grab sorc shoes to add a little more pep to your passive damage.

Overall advice

A lot of people will tell you to buy tear of the goddess on both forms. You can, its very good to do this, but its not so good as to be mandatory so feel free not to if you don't like it. I didnt before s11. Kayn is weak early, but a LOT of people underestimate him. If you early back and grab a long sword or two, you win 90% of the early jungle fights due to long sword difference. Even without them, if you are good at healing with E in fights, and canceling your Q you still do a LOT of damage very fast. And its hard to run from you or catch you. So feel free to invade if you are confident the enemy team wont rotate onto you. Blue buff is THE MOST POWERFUL BUFF on kayn, unlike most junglers where it is red buff. For Kayn, he gets the effects of blue buff, but due to the extra mana, he can spam E to heal like crazy basically giving him the effects of red buff as well. There is almost(almost. There are, but they are rare.) No circumstances where you should give your blue away. If blue gets invaded and stolen its not the end of the world, but it really sucks. You can solo dragon at level 5 or 6, but it feels better to get help. Make sure you have very good vision tho. I'd recommend buying a red trinket on your 2nd or 3rd back and picking up a control ward just before you head to dragon. You can also use Q to gain a little extra boost in walls. Imaging you are at dragon pit, in the wall, and you want to cross the river to get to blue side. You only have tier 1 boots. You can get close but shadow step times out before you make it. If you time it right you can use Q to close the gap and get inside the wall in time to not put shadow step on cool down. You can also use rift herald inside a wall. When farming or fighting I would highly recommend using your Q to dash straight into a wall, as that will make the Q animation end almost instantly and you can go back to autoattacking as fast as possible. Be careful to do this to a thick wall. To thin a wall and you will just dash OVER it, instead of hitting it and stopping.

Runes

Kayn is very rune fluid, a lot of things work on him. Phase Rush, Predator, Electrocute, so on. However by far Conqueror for red form, and Dark Harvest for blue for are the best by a mile. It gives you such good sustain and boosts your damage on red form, and very high burst and decent scaling on blue form. The smaller runes and off tree I'm sure you can work out on your own. If you still want my opinion then ask and you shall receive lol. To answer a question about the wrong runes, conqueror SUCKS on blue form, but dark harvest is alright on red form. Electrocute is typically what you run if you dont know what form to take, cuz it works for both. But like I said, 99% of the time you will know your form in lobby and just take conqueror or dark harvest.

Powerspikes and Items

Kayn has 3 distinct powerspikes for each form. For Rhaast its 1) getting your ult, 2) getting red form, and 3) getting goredrinker. Rhaast will always hurt and be hard to hurt, but he doesnt really do enough damage until you grab cleaver 2nd or 3rd item(2nd normally but 3rd if going muramana). Then he is a monster. After that just grab spirit visage, or a GA, maybe a deaths dance, or whatever you need. After cleaver you don't really need another damage item. As for Shadow Assassin, your powerspikes and items look like this. 1) getting ult, 2) getting blue form, and 3) getting whatever mythic you wanna rush. Duskblade, Eclipse, prowlers claw and goredrinker all have good points to be debated about. Then you can explode any squishy champion in the game if they dont dodge your spells or something. Assumeing your not like 2 levels down of course. After your mythic, grab a edge of night, or if you are really fed then maybe a Youmuu's(always get a lethality item before muramana if you choose to build one. If you run a lethality mythic then cool, but if you rush goredrinker, grab a edge of night or youmuu's before the muramana). Build mostly lethality and raw damage, but if you need you can grab a GA and still be just fine. Remember, your ult resets your blue form passive. 3 seconds of insane damage just became 6.

Choosing form

You should go whatever form you think you need. My personal rule is 3 or more squishes? Time for blue! 1 or less squishy? Time for red. Unless they have a galio or something like it. Then always red. Vayne WILL piss you off. You pop out of a wall to murder her and she instantly condemns you into the wall you just left.

Last words

Always go Chilling Smite. You have very little CC in your kit as Kayn and so the slow off blue smite is very good. It also is instant damage, at range. That's good for getting your ult off. Imagine they dodge your W, and your low or just need to chase them or something. You can smite them with blue smite to instantly ult. Red smite is nice on red form but blue smite is overall better for both forms. I think thats all the good information I have for a beginner. Any more questions please ask me! Good luck peoples!