https://na.op.gg/summoner/userName=vbniya
https://lan.op.gg/summoner/userName=xelniya
I've gotten to Gold twice using Kayle as one of my mains.
Here's my experience with how to build her. I've tried several builds with her and learned a couple of interesting things about how she scales. So lets talk about my most successful builds.
Her second strongest build in my opinion is this one:-

I get Tear as early as possible by starting
and then I rush
. You get it fully upgraded by 20 to 23 minutes if all goes well. From there I can get Blade or Runaan's. Blade is the stronger spike for fighting but Runaan's gives insane wave clear. So much so that you can rack up very high CS numbers quite easily and you basically one shot the jungle.
At full build you just melt everything in sight. As a matter of fact, at level 16 and up you can just wait for a team fight to start, ult into the middle of everyone and just instagib them, tanks and all. I once face tanked an entire team with my ult. I mean straight 1v5, no team mates around and got a penta with that build. It's main weakness is that it takes a while to come on-line. You won't really be a threat until Manamune is upgraded and your Blade is completed. It's a very passive early game build and don't expect to be able to do anything if you're team is hard feeding early game. A lot of the time you come online too late to do anything about your shit ass team mates feeding.
Another build I've used a lot is a variation of that first build.

Instead of rushing Muramana, you rush Blade as your first item. I've found that Blade of the Ruined King is the strongest possible first item spike you can get on Kayle. You can straight up start killing your opponent in a lot of match-ups when you start Blade. Works especially well vs Bruisers like Garen, Darius, Fiora and Olaf. At full build it's incredibly good at kiting due to amount of movement speed you get from the zeal items and your passive. No one gets away from you at level 16 with this build completed. And no one can catch you. The problem I've found with this build is that it's damage is quite low. I mean you're still Kayle but you're gonna struggle hard of they have a beefy team.
The strongest build I've found on her so far is this one:-

I used this one very briefly and for a handful of games. I went ER, Greaves, RH, then IE and a varying orders of the final items. The problem with this build is that it takes way too long to come online but if you manage to get to 5 items, it's gg. The damage is ridiculous. It suffers a bit vs beefy teams too but is better than the first build vs more squishy teams.
And the build I'm using currently:-

The main draw for me about this build is the 40% CDR which is very noticable. You always have your ult up with this build. You can start Ludens, Nashors or Rageblade fitting in Greaves when you feel you can. Another advantage of this build is that it takes advantage of your AP scalings so your heal and your waves benefit from it more so than any other build. The main strength of this build is that you can start helping you team very early. Always having you ult up is a big part of that but also, the AP gives you incredible burst. It also scales really well into the late game. It's greatest weakness is that it's terrible at dealing with tanky teams. The other builds are a lot better at dealing with tanky teams. I use this build currently mainly because of the 40% CDR. It's really really fuckin helpful having your ult up all the time. 40% CDR feels too good on her to pass up. However, if they were ever reduce her ult cooldowns, I would default to first or second build because they are quite frankly, more powerful builds.
One last thing. There is a popular full AP build where you take Transendance and Gathering Storm then build as much AP and CDR as possible. I've seen videos of guys getting 1000 AP with this setup and just destroying everything in sight. It may even be her strongest build, I'm not sure. I don't like it though because you lack attack speed which feels painful to me. You can also try that one if you want.