The way I have been playing Kayle is a AP/AD hybrid with good attack speed that can take out teams when the build is done. This build is extremely item dependent. However when you get a couple of items the pieces start to fall in place.

Going for Heavy speed,
-AP so your judgement will be feared.
-AD so even if they get MR you will still do damage.
-Life-steal and Spell Vamp is the icing on the cake, having splash spell vamp is nice.

I prefer to have Ghost, it can be easily replaced with clairvoyance and exhaust.

Positive Spells:
-Clairvoyance, Good to use early game to snipe champs out of the bush, and when Champs run away and hide in bush waiting for a jump you can snipe them out.
-Teleport, Always good to have, can go back heal and shop then resume the farm.
-Flash, Obviously
-Ghost, Good to have when on a chase with Kayle's slow. Also can come into handy when an ally is in trouble and you can reach them in time to Heal/Shield them.
-Clense, Good to use because Kayle's Ulti is still effected by stun


Neutral Spells:
-Ignite, Not a big fan of it, only really good if you want to get early kills with Kayle which is not easy to do but not hard.
-Exhaust, Kayle already has a slow that boosts her damage by 5%. Matters on preference.
-Fortify, Good if facing a team of turret divers. Shield up, kite then let tower kill.


Negative Spells-
Heal- No. just no
Clarity- Good early game then useless late game.

Pros:
-Good at Lane Pushing
-Good Hero Killer
-Good Semi-Support

Cons:
-Will get Focused in both
-Team Fights
-Early Game

As Kayle dps Hybrid that main this that you should be concerned about is staying in the back of the group waiting for your team to start the fight. When the fight starts be considerate of your Ulti.

What not to do:
-Run in front to chase an enemy when there still is a group of people. You will get focused and force to use your ulti to run away.
-If your damage output is better then others don't put shield on them.

What to do:
-Focus on the squichies, if they overextend destroy them. Hardcore.
-Use your Splash to your advantage, if a Champ is too far away aim close to them to still do damage.
-Use Heal when chasing/For tank have to eye those things out when is a good time to use it and not.

Kayle's splash is Epic. A few hits and massive creep waves are dead. Always those guys who try and run over and steal your waves. Not cool Bro. I suggest go into you jungle on a route from lane to lane killing. Works for the most part. If you are losing bad then it will be hard though.

Kayle Can dish out the damage, save her allies and just destroy any creep or tower that comes her way.