Who should I buy,Yasuo or Kayn?

Depends if you want to jungle or if you want to lane.

If you would like to play jungle or sometimes top, buy Kayn. He can have either a lot of burst or sustain depending on enemy team composition. Can be fun when perfected. Best at taking out high priority targets before they deal damage to your team, especially as shadow assassin. Good at taking damage and dealing damage back against tanks and adc's as Darkin.

If you love to be hated like Satan himself, buy Yasuo. He has a lot of mechanics and while his basics are rather easy, his advanced techniques and knowledge can be hard and time-consuming, but very rewarding. A basic-oriented champion, his kit does require putting in as much aa's as quickly as possible in between his abilities, which many people don't realize. Also, position is important as his kit requires enemies for max mobility. Good at blocking damage from enemy adc's, extending cc, and deals a lot of damage to high priority champions if he manages to reach them.

I know a lot about Yasuo (love the hate, it's an acquired taste []), but only know how to theoretically play Kayn (only about 10 games). They are both fun, but have different playstyles.

neither! >:( they are both edgelords, thats the equivalent of asking if you should be a Reaper main in Overwatch. buy or instead. Kha moves at 350 base MS, and Yi moves at 355. Yi can just ult after people if they try to escape, Kha can jump over walls or turn invisible.

also i downvoted because of your desire to play yasuo