Kayle is a champion that almost no one plays and thats a shame. She is really fun and also very strong. And most of all, she is easy to play. Building her with attack speed and on-hit effect items, is what i found best. I've tried her aswell as AP and AD, but didnt found that usefull.
She is really fun and you can almost 1 vs 1 every champion! She is great 1 vs 1 with carries.

I recommend Berserker's Greaves, Phantom Dancer, Malady, Wit's End and Black Cleaver, for attack speed, attack/spell damage, armor pen, magic res and on-hit effects.

Important: I would say, always buy items with attack speed! At the end, it makes you incredible fast.

But buying Madred's Bloodrazor or Ionic Spark depends on enemy team. Madred's Bloodrazor should you buy if enemy team is strong on HP, while Ionic Spark is great for fast pushing / ninjapushing, but also very usefull in teamfights aswell, if your team have any kind of aoe stun or snares, like Galio, Amumu and Leonas ultimates.

I recommend Armor Penetraion marks, Armor seals and Magic resist pr. level glyphs (standard carry runes). Armor and magic resist is usefull fighting 1vs1, aswell with the armor pen.

The Quintessences are your choise. I use armor penetration, but attack speed or movement speed is great aswell.

For summoner spells you can use whatever you want, i just preffer Heal for 1v1 and Exhaust or Ignite. Exhaust for easy fighting carries and ignite for kill-secure.

Flash, Ghost, Cleanse, Surge, Teleport would also be usefull, and Clarity aswell if you are having mana problems.

I always play Kayle on botlane with some1. Anything but carries is usefull, since they need farm and you do aswell. I find laning with supports or fighters great.

Like said above, you need farm as Kayle. You dont need carry-farm, but still need some to get your items going, else you aint much worth. So if you lane with some1, you can share farm.