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
Master Yi, AP
Champions that were player skill dependent
Champions I could not beat
Jayce, ( That ' To the Skys! ' )
Talon, (except this one Talon that was really readable when he was going to use Rake)
* 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.