Whatever you have heard about AP Kayle, disregard it. AP Kayle is just as viable as AD or Hybrid Kayle for DPSing. AP Kayle starts off as a nice support character with good harassing capabilities and quickly turns into a damage-dealing machine, capable of tearing apart enemies very fast. This build goes mostly AP (With a little AD thrown in for good measure) to increase the damage dealt by Reckoning and Righteous Fury. Since this build focuses on abilities, cooldown reduction is a must. (As a side note, this build is also pretty effective in ARAM if you get stuck with Kayle)
I go for a 21/0/9. Being the versatile champion that she is, Kayle can also go 9/21/0 (which would be the best alternative), 0/21/9, or even 9/0/21. All are viable and depends on preference.
Holy Fervor - 30% of Kayle's attack damage is added to her ability power and 15% of Kayle's ability power is added to her attack damage.
This is the best thing that ever happened to Kayle. Riot probably noticed her use was dying down and decided she needed an upgrade. Well, an upgrade is definitely what she got.
Reckoning - Blasts an enemy unit with angelic force, dealing damage and slowing movement speed. While the target is slowed, Kayle inflicts increased damage to them.
Blasts a target, dealing 60/110/160/210/260 (+0.8) magic damage and slowing its movement speed by 25/25/25/25/25% for 4 seconds.
While the target is slowed, Kayle inflicts 8/10/12/14/16% increased damage to them.
Your initiate/harass skill, and a fine one at that. Deals a significant chunk damage with all of your AP. This should always be on your target to get the increased damage from it.
Divine Blessing - Blesses a target friendly champion, instantly healing them and granting increased movement speed.
Blesses an allied champion, healing them for 65/100/135/170/205 (+0.5) health and increasing their movement speed by 12/12/12/12/12% for 10 seconds.
Not the best healing spell in the game but still fairly significant. The speed boost is the star here, great for getting around or helping a fleeing ally. Use this as often as you can.
Righteous Fury - Ignites Kayle's sword with a holy flame, granting Kayle a ranged splash attack and bonus magic damage.
Kayle harnesses her righteous fury to increase her attack range by 400 for 10 seconds. These attacks splash, dealing an additional magic damage to her target and nearby units take 20/25/30/35.5/40% of her attack damage plus magic damage.
The total bonus damage is 4/10/16/22/28 base damage plus (+0.3) (30% of her ability power).
This is what makes AP Kayle go from alright to "OH NOES". Significantly increases your damage output (we're talkin' in the hundreds) and makes farming easier. A must for any battle situation.
Intervention - Shields Kayle or an ally for a short time, causing them to be immune to damage.
Bathes Kayle's target in a holy light, rendering them immune to all damage for 2/2.5/3 seconds.
A cool thing to have, but not super awesome. Hard to pull off, but in the right situation it can change a fight or save someone's ***. Don't limit it to yourself, use it on a dying/beefy ally.
Overall, Kayle's abilities make her a great DPSer, but you need some CDR to make it work.
UPDATED! Thanks to Searz and Tinaby for their contributions!
Banshee's Veil is your NEW starting item goal. Upon testing, I found it to be FAR superior to Rod of Ages. Start off with a Sapphire Crystal and a few Health Potions. Get Catalyst the Protector as fast as possible to help you stay in the lane longer. Like I said earlier, CC can be a real problem for using your ult. With Banshee's Veil and Mercury's Treads, CC shouldn't be a problem. If you feel confident, though, get Boots of Swiftness. After that, you'll get a Nashor's Tooth. Nashor's Tooth is probably the best item for AP Kayle ever. +55 AP, +50% Atk Spd, 25% Cooldown Reduction and Mana Regen to boot. All vital for AP Kayle and in one item! After that, I get a Lich Bane mainly for the proc and movement speed. The mana and MR is great, too. Hextech Gunblade will be the next item, for the great boost to DPS and the lifesteal/spell vamp. Afterwards, Rabadon's Deathcap will give you a SERIOUS increase to AP. If you life the extra armor or the active you can take Zhonya's Hourglass instead. Other options include Rylai's Crystal Scepter for Health and CC and Archangel's Staff if, for some reason, you want more mana (although with Banshee's Veil, it might be pretty good). Deathfire Grasp is also a great option for putting a serious dent in a tank's health (thanks cpen). The new Malady would be a pretty good thing for Kayle. I don't know what I would replace with it, maybe Lich Bane.
Kayle is fairly squishy early on, so be careful. Once you get Catalyst the Protector, you can be a little more reckless. Harassing is your forte. Reckoning and Righteous Fury can dish out a lot of ranged damaged if used consecutively. Watch the timer on Righteous Fury, though. When it runs out, you will go towards them for melee. Use Righteous Fury to farm and you will have a Banshee's Veil before long.
At this point you should have Banshee's Veil and boots, so you can start moving around a bit more. Help your teammates, gank if you can (probably not alone) and farm often.
Once you have your Nashor's Tooth, Lich Bane and Hextech Gunblade, you will become a powerhouse of damage. Pick a target, hit Reckoning, then Righteous Fury. Hit Reckoning as often as you can for the damage increase and the slow.
So there you have it.
Try it, rate, comment. I hope Kayle catches on more. She's an underrated champion and deserves a second look.