Forewarning, I've only manage to play Kayn once in both forms, so don't consider me an expert. I have been wrong about champions in the past. These are just my personal thoughts based on playing as, playing with, facing against, and hearing about Kayn.
First I want to talk about his design. Lots of people have dubbed him to be "edgy". Urban Dictionary's top definition is "Taking coolness to its extreme and generally beyond the realm of actual possibility, while at the same time seemingly unaware of how ridiculous it is." I mean look at Kayn. He's a ninja, he has a demon scythe, he has cocky personality, he has anime hair. League of Legends is filled with a huge variety of champions, so I'm pretty fine with the design. Unfortunately, the problem with using tropes is that the character feels kind of bland, which unfortunately I feel Kayn is in terms of personality and backstory. The most interesting part of Kayn is his other half Rhaast, and how Kayn interacts with him. While it's not necessary to have to put a twist on every single trope to spice it up, considering the higher standard being placed on trying to keep champions from being "one dimensional", Kayn has missed the mark.
On to gameplay thoughts. I must admit Kayn has been disappointing. There was a term that I remember someone on Reddit described called "power fantasy". Basically, it's how you think the champion should feel in terms of its design. On paper, Kayn looks like he'd do lots of damage, unfortunately, he doesn't feel that way unless he's ahead. His abilities just doesn't feel impactful in both animation and damage. While it's understandable he's a champion that is suppose to scale with his transformation, ramping up to that state just seems really difficult. His passive looks to reward aggression and ganking, but his damage is just really poor and kind of difficult to land with his short range and lack of CC. He just seems far to reliant on his teammates to succeed to succeed himself. I mean look at the role Kayn basically fills. It's either assasssin or diver. Both which have a rather poor history of being feast or famine.
In general, this champion feels incomplete in execution. I know this review is rather incomplete in itself as it's pretty much just criticism without details or solutions, but I had to get these thoughts out before his release even if he's more than likely to be shipped in his current form.