When to go shadow assassin and when darkin kayn?

Ask your team and then mute all and do the exact opposite of what they say

As soon as you get it USE IT and if they have tanks / jugs go RED and if they have more like SQUISHY go blue

Assassin Kayn is ideal if you're facing a squishy team.

Darkin Kayn/Rhaast is often taken because he's a bruiser that heals through his abilities : he can sustain a long time in teamfights. He's overall better if the enemy team has a tank, since he will be "tanky" because he won't die easily and he will also deal damage. ( he's also built as such )

Basically, Shadow assassin is super niche. Only take it if the entire enemy team is squishy. Rhaast is a much safer pick almost every game simply because he works against almost everyone. Incredible sustain in fights, decent burst damage, and overall just a better pick. However, if there isnt a tank on the enemy team of any sort, go SA and slaughter them.