Kayn is a very unique champ. The most unique champion in the game to be exact. Why? Well, for a couple of reasons. First off, he's the only champion in the game who can actually walk through walls. Not to mention he's one of the most mobile champs in the game because aside from Lee Sin he's the only champion who has two ways of moving past terrain without the need of an ult. However, unlike Lee Sin, our boy Kayn is free to move past terrain freely, unlike Lee who needs to land a skillshot. Either that or have an ally or ward nearby. And to top it all off, Kayn is the only champions who can change his class in the middle of a game. So why should you play Kayn? Because, he's secretly op. Judging by this statistics, Kayn is one of the worst junglers in the game as of now, but truth is, he's actually incredibly powerful. The only problem is that most people don't know how to build him, or let alone play him correctly. With that in mind, before I explain further, let's take a quick look at Kanye's strengths and weaknesses as well as the proper runes and masteries for him.

Pros

S tier mobility
One of the best invaders in the game
Has very unique gank pathing thanks to his E
Can change his class depending on what your team needs
Is insanely powerful once he undergoes a transformation

Cons

Is very weak in his base form
Has subpar ganks due to his lack of damage and cc.
Is a feast or famine champion, meaning that sometimes you'll get fed to the point of being unstoppable, but at other times fall so far behind that you'll practically be useless.

For my runes I go for flat armor seals, flat MR glyphs, and full lethality marks and quints. As for my masteries, I like to go 12/18/0 with Thunderlord's decree as my keystone, unlike most Kayn players who run AD marks, AS quints, and Fervor of Battle, which makes absolutely no sense whatsoever. This is considered to be the standard set up for Kayn, but think of it this way: Kayn is a bursty caster, not an auto-attacker. With that said, it makes a lot more sense to capitalize on his burst rather than his sustained damage. And yes, this is the setup that I use for both Shadow Assassin AND Darkin Kayn. Now, if you know for a fact that you're going to go Darkin, like you are hell-bent on going Rhaast, and you will wait 4 minutes if you have to, then you can opt for a tankier build with Grasp of the Undying as your keystone, but I would not recommend this, since doing so will make you lose a significant amount of damage, which isn't good considering that Rhaast is more of a diver than he is a true front liner. Sure, he isn't nearly as bursty as Shadow Kayn, but usually you'll still be focusing the enemy carries whenever the opportunity presents itself, and THEN transition to the enemy tanks and bruisers, which Rhaast can kill fairly easily thanks to his massive percentage health damage.

If you don't know what Kayn's abilities do, then I would strongly suggest watching Kayn's champion spotlight. Here's a link to it: https://www.youtube.com/watch?v=pPaauwI6G5s

When playing Kayn in his base form, it's important to not only gank whenever the opportunity presents itself, but also to invade the enemy jungler whenever you get the chance to do so. Now, unless the enemy jungler is low on health or OOM, I wouldn't suggest going for any trades, since you'll most likely loose. If anything it's best to get in, and get out, which Kayn is great at thanks to his S tier mobility. Normally, I start Raptors, then transition to my Red Buff, Wolves, Blue, and then I go to the enemy jungler's Krugs. Now, when ganking, it's not only important to know who to gank, but also when to gank. The reason I say this is because Kayn gets essence from attacking champions. Attacking ranged champions will put points into Kayn, whereas attacking melee champions will put points into Rhaast. Additionally, the value of these orbs will increase as the game progresses, meaning that you won't necessarily have to camp the same champion in order to go into your desired form. Now, when you do unlock a form, you have the option of going into that form immediately, or you can unlock the losing form 4 minutes later. Typically though, I would recommend unlocking the form you get first, because 4 minutes is an awfully long time to wait for a crucial power spike. That, and the form you unlock first is usually going to be the form your team needs anyway, even if it may not look it.

You have ousted the Darkin, Rhaast, and are now ready to kill your enemies silently. Or at least, on your end. On the enemy ADC's end, they're probably going to be screaming, slamming their keyboard, and punching their monitor, because HOLY URF does this guy do damage! Now, your teammates might scratch their heads since Rhaast is generally considered to be superior, but that couldn't be farther from the truth. If anything, Shadow Assassin Kayn is extremely deadly thanks to his passive, which gives him what is quite honestly the highest burst damage of any AD assassin. Remember old Talon and old Rengar? Yeah, that's how bursty Shadow Kayn is. If you're super ahead you can one shot a squishy with just your W and Q alone. It's disgusting. Now then, how should you play Shadow Kayn? Well, as the name implies, like an assassin! Simply look for out of position squishies, spook the Teemo shrooms out of em, and laugh as they futilely attempt to flee. As for teamfighting, make sure that you choose the right time to go in. If you go in instantly, they'll just surround you, and pop you the moment you emerge from your target. But, if you're patient and choose your moments wisely, you'll be making some big boy plays in no time.

Congratulations! You have pierced Kayn through the chest with your unholy scythe, and have now entered your "I just want an Aatrox rework" form. As Rhaast, your sustain rises drastically, and a lot of your flat damage gets replaced by percentage max health damage. Time to chomp on those tanks, right? Well, sometimes. Not always though. If anything, Rhaast is more of a diver than a true frontliner, so don't try to start teamfights yourself. If anything let your teammates engage, and follow up on top of their cc. CC heavy champs such as Alistar, Thresh, and Ornn make for excellent teammates for Rhaast. BUT if the opportunity presents itself, try to knock up squishies with your W, and go onto them. If you can kill them, great, but if not, just try to buy as much time as you possibly can. Rhaast is great at this thanks to his unprecedented amount of sustain and disruption. Now, do keep in mind that sometimes this will get you killed, but that's just the reality of being a front liner. If you play your cards correctly though, you should be able to Shadow Step to safety, and if not at least you enabled your team to clean up. So in the end, just get in there and start slaughtering everything you can get ahold of! I'm sure the enemy team won't mind.

I hope this guide has given you some insight as to how to properly play Kayn. If you play your cards right, your match history should look somewhat like this:
(For some reason I couldn't upload these images directly to Mobafire. Every time I tried I would get a box with an X on it instead, so until I find a hotfix, here are the links to the images).

Shadow Assassin:
http://imgur.com/u0sZSuk
http://imgur.com/eGYTfke
http://imgur.com/tAoj8dv

Darkin:
http://imgur.com/lLynO6d
http://imgur.com/fYBGvYt