Heya people!
A little bit about me : I’m best known for being a Talon one trick and if you would like to improve at him too you can check my other guide for Talon. ^^
I have watched Kayn progress since PBE and have been a huge fan of him and I’m looking to second main him , so I decided I’d make a guide , hope it will help you!

Kayn's Passive

At one point in the game , Kayn will permanently transform into either a Shadow Assassin or a Darkin.

Kayn's Q

Kayn dashes , then slashes. Both deal damage.

Kayn's W

Kayn damages and slows targets in a line.

Kayn's E

Kayn can walk through terrain.

Kayn's R

Kayn hides in an enemy's body , dealing massive damage when he bursts out.

I chose Reaping Slash as my first ability for early AoE damage to boost your already amazing clear and flexibility with ganks or clear paths with the dash part.
I went for Blade's Reach second for AoE clear and the slow which helps in kiting and ganking.
Shadow Step third because even though it doesn’t provide damage it is your main ganking tool and helps regenerate health after a harsh clear. You can opt to put your fourth point into this and instead get Reaping Slash third for more damage if you don’t plan on ganking.

Flash – Probably the best Summoner Spell in the game. You can use it in your combos and for escapes , invaluable to Kayn.

Smite – Well you are going to be jungling most time you’ll play Kayn , pretty self explanatory.

Other options:

Ghost – It is a strong spell and helps you chase a lot better , but in my opinion Flash is just a safer and maybe even more rewarding spell.

Ignite – If you are going Mid lane or Top lane you are obviously not taking smite so this is a very good choice.

Teleport – If you are going Top lane and will not play aggressive or the enemy team has a lot of pressure with their Teleport this is the preferred option.

Shadow Assassin

Pros:

-Very high burst damage
-High mobility and map presence
-Invulnerability from Umbral Trespass
-A lot of escape potential
-Unpredictable engage/flank patterns

Cons:

-Squishy
-Scales off late game
-Vulnerable to CC and peel

Darkin

Pros:

-Good frontline bruiser
-A lot of sustain
-Hard CC (Knockup) from Blade's Reach
-High dueling potential
-Invulnerability from Umbral Trespass

Cons:

-Can’t reliably replace a Tanky engage frontline
-Predictable ability patterns , especially for your only form of CC Blade's Reach
-Can’t dive backlines very well without getting chunked very hard

Marks - Full AD runes , can alternatively be mixed with Lethality for better scaling.I have seen people running both full AD and full Lethality on these , but I believe a mix of them gives the best results. AD is very good for pushing towers , clearing the jungle and scales better early. Lethality helps you transition from early to mid game smoother , since Lethality outscales AD as the game progresses.

Seals - Armor , if you don't need Armor and are against more AP champions or just simply look to scale better late you should get Scaling Health instead.

Glyphs - Magic Resist.

Quintessences - AD , it helps with your clear and pushing towers. I have seen people get Lethality Quints but Ad Quints outweigh their damage early game and since your early game is a bit weak you need them.

Shadow Assassin

You can either go for Stormraider's Surge for extra stickiness to your target and escaping after successful kills or for Thunderlord's Decree for more burst.
Going by personal preference and overall more useful pick considering Kayn’s kit I would go for Stormraider's Surge since the extra damage from Thunderlord's Decree isn’t going to do much in late game teamfights.

Darkin

Here you have multiple choices depending on the game : Stormraider's Surge , Fervor of Battle and Courage of the Colossus .
You can pick Stormraider's Surge if you are not sure what form you are going for for middle ground , although it doesn’t synergize as well with Darkin’s kit.
Fervor of Battle will be your pimary choice because you’re going to duel and teamfight a lot and this will boost your damage significantly in prolonged fights.
Courage of the Colossus should be picked if you are your team’s only frontline and need the shield on engages. It is a good keystone if you can reliably hit your knock up in teamfights.

Shadow Assassin

Stalker's Blade - Warrior is what you'll be going for most of the times because it helps with chasing or escaping and grants more stickiness.
Youmuu's Ghostblade is a very good item to help with your ganking and preferably not delayed much.
Duskblade of Draktharr is another strong item that helps early-mid game since not a lot of people get to stack armor by then so you’ll make use of it’s passive a lot , also the slow , while not very significant is strong in ganking.
Black Cleaver is a really solid item overall , that's the reason you'll be getting it for both forms , it has defensive stats , offensive stats , armor shred and grants stickiness to your target thanks to its passive.

Darkin

Stalker's Blade - Warrior is what you'll be going for most of the times because it helps with chasing or escaping and grants more stickiness.
Spirit Visage is really good and part of your core because it amplifies what the Darkin revolves around - lifesteal.
Black Cleaver is a really solid item overall , that's the reason you'll be getting it for both forms , it has defensive stats , offensive stats , armor shred and grants stickiness to your target thanks to its passive.

As Kayn your combos are pretty straightforward :

1) Reaping Slash > Blade's Reach

2) Blade's Reach > Reaping Slash

3)(Surprise Attack) Shadow Step into terrain > 1) or 2)

Your Umbral Trespass is not used by default in your combos as you need to decide in combat when and for what you are going to use it , a bit of invulnerability after diving , finishing off targets or escaping.

Your early game clear is good , by the end of it you should be around half health or more.

In the early game , you should look to gank a lot , and also take into consideration ganking lanes that will give souls for your desired transformation.

In ganking , you should almost always have your Shadow Step available and preferably your Smite with the Stalker's enchant. That is because your slow from Blade's Reach often isn't enough to secure a kill so you need the extra stickiness from Smite and great positioning from Shadow Step.

By mid game you should have transformed , otherwise you'll get behind easily and punished hard.
In the mid game you should be looking to group with your team with objectives and farm when possible , your items are strong power spikes.

Try to scout and ward their jungle when you can and predict their jungler's movements. Mid game is where Shadow Assassin shines , so make use of its power.

For late game you should try to catch lone targets and get picks , group up with your team and focus on objectives.

Sadly , if you play Shadow Assassin you're going to scale off a bit late game so you should try to end the game as fast as possible , although if you are Darkin late game is where you scale the best.

Thanks for reading guys. :)

Be sure to leave some feedback and suggestion for my guide , and maybe take a peek at my other guide for Talon! ^^