Over the course of my 2 weeks or so playing Kayle bottom lane, heres what I can remember for my match ups.
Champions I could beat
Akali
Annie
Caitlyn
Draven
Ezreal
Jax
Kog'Maw, AP
Master Yi, AP
Miss Fortune
Morgana
Mordakaiser
Orianna
Ryze
Swain
Syndra
Varus
Zed
Champions that were player skill dependent
Brand
Cho'Gath
Heimerdinger
Katarina *
Kayle
Lux
Maokai
Pantheon
Singed
Teemo, AD
Vayne
Xin Zhao
Champions I could not beat
Ashe
Darius
Elise
Fiddlesticks
Fizz
Garen
Jarven IV
Jayce, ( That ' To the Skys! ' )
Kassadin
Kha'Zix
Leblanc
Malphite
Nunu
Renekton
Talon, (except this one Talon that was really readable when he was going to use Rake)
Teemo, AP
Vi
Zilean
* The thing about Katarina is that both Champions have enough damage to kill each other in one go. Its a fight of who uses their ultimate first. If Katarina uses her ultimate first, then Kayle can shield herself and win. If Katarina makes Kayle panic and shield herself early, then Katarina just needs to wait it out and then kill Kayle. I've beaten and lost to Katarinas in both lanes and jungle off this simple principle.
Now of course this is just my memory of my match ups which is not perfect and each match up was my skill level against theres so this is not set in stone. I'm certain someone else can beat the champions I cant with kayle and vise versa.