Kayle is not a very frequently used champion. She's one of the cheapest champions, and I guess it makes people think she sucks. But look at Master Yi. He can kick your butt - well, slice it in half - if the player knows what he's doing. Kayle can be powerful too.
In this guide I will teach you how to play Kayle as an Attack Speed/Ability Power Hybrid. At first, it may sound weird. But Kayle's Righteous Fury allows her to auto attack from range and deal bonus Magic Damage on each hit. It also grants her auto attacks a splash damage effect. The damage is increased by Ability Power and Attack Speed will let her land more hits during the skill's duration. Combining these stats makes Kayle extremely powerful from range.

This build will make Kayle a strong carry, and is most effective if you're on on mid or solo top!

And, hey, I'm not an expert, but this build works very, VERY well for me.
Enjoy! :D

Pros

• + Great support
• + Great damage (AoE, too)
• + Difficult to take down if abilities are timed correctly
• + VERY easy to farm
• + Not very hard to play

Cons

• - Mana consuming during early game
• - Fragile during early game

I'd suggest to have one offensive and one defensive summoner spell, though the defensive spell may also be used for offensive tasks and vice versa. I'll explain below.

Slot 1 - Offense

Surge

Surge is the primary summoner spell in this guide. It grants Kayle both Attack Speed and Ability Power - Exactly what we want! This spell will help you get some early-game kills, and it won't lose it's effectiveness as the battle progresses, because the Ability Power bonus increases with each level. This spell allows some burst damage to be done!
While Surge by far is the best choice, you CAN use other spells, if you really want to:

Exhaust

Exhaust is a spell that can be both offensive and defensive. It's good when chasing, it's good when being chased. Not only does it reduce the Movement Speed of the enemy, it also decreases it's damage done. The reason I would consider it an offensive spell is that it greatly increases your chance to win a 1v1 - especially if you picked the Mastery Summoner's Wrath .

Ignite

Ignite does quite some damage - TRUE damage, that is, meaning it ignores Armor and Magic Resistance - something we can't say about Surge. Also, if you follow my instructions on Masteries, you gain 5 Ability Power and Attack Damage when Ignite is on cooldown. It isn't much, but it's definitely a plus.

Slot 2 - Defense

Ghost

Ghost is one of the spells that can be both offensive and defensive. Just like Exhaust it's good when chasing and when being chased. Summoner's Wrath increases the Movement Speed bonus.

Flash

Flash can be useful in many situations. It CAN be offensive, but I think it's a waste to cast it to get one kill, if it could've saved your butt a minute later. If you rely on Flash and always expect to be able to escape because of it, it should NOT be used if not absolutely necessary. Flash has a longer cooldown than Ghost and that is the main reason I prefer Ghost. You can't escape with Ghost if you get stunned, though.

Early game

• Start off with Boots and a few potions. I usually buy 2 Health Potions and a Mana Potion. Even though Kayle can heal, potions are necessary. Early in the game she has low Mana and you don't want to waste it all on healing wounds, that could've been healed by a potion. I'm not saying that healing is a bad idea - I choose the skill at level 2 - it's jusy very mana consuming if cast it too frequently.
  
  
• Early in the game, Attack Speed is more important than Ability Power. Therefore upgrading your boots to Berserker's Greaves is a good idea.
  
  
• Now it's time for some of those Attack Speed/Ability Power Hybrid items. Buy an Amplifying Tome and upgrade it to Malady. Malady's Unique Passive makes your Righteous Fury pretty painful - and Kayle's passive ability Holy Fervor already decreases enemies' Magic Resistance. Note that it does not only increase YOUR damage to the target. If your allies deal magic damage to the same target they will also deal more damage.
  
  
  Alternatively, you can buy Wit's End instead. It almost grants you the same Attack Speed bonus and it deals more than twice as much on-hit Magic Damage as Malady does. Also, it grants you Magic Resistance (and an on-hit Magic Resistance effect). It does not give you Ability Power, though, meaning that your Divine Blessing and the splash effect from Righteous Fury won't benefit from this item. Wit's End should be bought if the enemy team is building Ability Power, as it can protect you from their magic.
  
  

  Why not Guinsoo's Rageblade?

  I've been asked why I don't use Guinsoo's Rageblade in my build. The answer to this question is quite simple. First of all, it grants Attack Damage, a stat I am NOT focusing on in this build. It's not bad, though. The main reason why I'm using Malady and Nashor's Tooth is the Attack Speed bonus. Even when the Rageblade is fully stacked, it's Attack Speed bonus is not as effective as the bonuses from Malady and Nashor's Tooth.
  Thanks to DarkTank2 for the question.

Mid game

• Now when you've got some Attack Speed, you'll want some Ability Power. Buy a Fiendish Codex and upgrade it to Nashor's Tooth - another Attack Speed/Ability Power Hybrid item. Along with some Ability Power and Attack Speed it also increases your Mana Regeneration and Cooldown Reduction. From now on your Mana should be fine. The Cooldown Reduction is also very important as it lets you activate Righteous Fury (and other abilities of course) more often. If you follow this guide, the cooldown on Righteous Fury should be around 9.65 seconds at level 18 - maybe sooner. The ability is active for 10 seconds and the mana cost is only 45, so you will be able to have it active at ALL times. Of course you're not supposed to if no enemies are near.
  
  
• Now it's time for some Ability Power! Rabadon's Deathcap! Enough said!

Late game

• Now, THIS item is awesome: Lich Bane! Ability Power, Magic Resistance, Mana, Movement Speed - all good stuff! But what makes it incredibly powerful is it's Unique Passive... After using an ability, your next physical attack will deal bonus Magic Damage equal to 100% of your Ability Power. Even though Righteous Fury deal bonus Magic Damage it is STILL a physical attack. This means that the first hit with it will hurt quite a lot.
  
  
  Alternatively, you could buy Trinity Force. Attack Damage, Ability Power, Attack Speed, Movement Speed, Mana, Health, on-hit slow effect. All good stuff! The main reason I prefer Lich Bane is that it's Unique Passive adds your Ability Power as bonus damage on your next attack. Trinity Force increases your Attack Damage by 150% for one attack. Using this build, you will get NO Attack Damage from items (unless you buy Hextech Gunblade as the last item (explained later) other than Trinity Force, if you decide to buy it. So the bonus damage effect will not be as effective. Also, Lich Bane grants you more Ability Power than Trinity Force AND gives you Magic Resistance.
  
  
• Last item... this one is a bit difficult. There are many options. I'm gonna go through some of them:
  
  

  Rylai's Crystal Scepter

  In general, this item is probably the best choice due to the on-hit slow effect ( Righteous Fury triggers the effect). It also grants you quite some Ability Power. It depends on the situation, though. This should be picked if you are dominating the battlefield and are tired of enemy champions who escape with, like, 10 Health left (if you decided to buy Trinity Force instead of Lich Bane you should not buy this item, because you already have an on-hit slow effect). REMEMBER: you can already slow enemies with Reckoning. The reason I like Rylai's Crystal Scepter is that Reckoning has a cooldown and cannot be spammed THAT much. Also, the AoE magic damage from Righteous Fury will slow enemies around your target by 15% if you choose to by the Scepter.
  
  
  

  Hextech Gunblade

  Even though I have not focused on Attack Damage in this guide, it's definitely not a bad thing! The Life Steal will not be VERY effective, but the Spell Vamp will make you hard to take down. Also, the Unique Active kinda makes up for the slow effect of Rylai's Crystal Scepter - but not entirely. You can consider it a second Reckoning. This items should be picked if you're having a hard time surviving ganks.
  
  

  Will of the Ancients

  Will of the Ancients is a great choice if many of your allies are buying Ability Power items. The aura grants both Ability Power and Spell Vamp. This means that you will be healed for a percentage of the Magic Damage you deal - and nearby allies get the same buff! This items is GREAT when combined with Malady and Holy Fervor!
  
  

  Rod of Ages

  Health, Mana, Ability Power. It's effectiveness increases with time. Should be picked if you dominate the battlefield, but always end up with low health after the ganks. It will make you more sustainable.
  
  

  Zhonya's Hourglass

  Rabadon's Deathcap is the only items in the game that grants you more Ability Power than Zhonya's Hourglass. Buying another Deathcap could be an option too, but the Unique Passive does NOT stack - keep that in mind. Therefore I think Zhonya's Hourglass is a better choice. The Unique Active is great in ganks. But use it wisely. You can not take ANY actions when it's active. You can't even move. This should be used if the enemies focus on your during ganks. It allows your allies to deal some damage while the enemies are wasting there time trying to kill you. Remember that Intervention can be used in the same way. The difference is that you can also cast it on allies AND the target is able to move and take actions. If you're in focus during ganks and you aren't happy with Intervention, Zhonya's Hourglass is the way to go.
  
  
  

  Void Staff and Abyssal Scepter

  These items are very similar to each other. They both grant you 70 Ability Power - so that's a tie. Void Staff gives you Magic Penetration and Abyssal Mask reduces the Magic Resistance of nearby enemies. Soo, what's the big difference? I'll tell ya: the Magic Penetration will probably make you hit a lot harder. The Magic Resistance Reduction Aura, though, will make you AND your allies hit harder - when dealing Magic Damage - because the effect does not apply to your attacks, but simply decreases enemies' Magic Resistance. YOU will hit hardest with Void Staff, but if some of your allies are bursting with magic damage Abyssal Mask would help the team much more. So I would probably go with Abyssal Mask. As a bonus you'll also get some Magic Resistance!
  
  

  Summary

  Item Sequence

  

  

  

  

  

  

  Health Potion 50

  

  Health Potion 50

  

  Mana Potion 0

  

  Berserker's Greaves 1100

  

  Amplifying Tome 400

  

  Malady 2035

  

  Fiendish Codex 900

  

  Nashor's Tooth 3000

  

  Needlessly Large Rod 1250

  

  Rabadon's Deathcap 3600

  

  Sheen 700

  

  Lich Bane 3100

  

  Rylai's Crystal Scepter 2600

There's not much to explain here. The build is about Ability Power and Attack Speed, so that's what you'll want. One thing will be important, though: If you prefer Flash from Ghost you'll probably want Summoner's Insight instead of Improved Recall .

Summary

While items are very important, Runes and Masteries are not. Well, of course they make a difference, but they are not as powerful as items are. Masteries are "free" provided your level is not too low. But Runes cost Influence Points. Therefore, you shouldn't buy runes specifically for one champion, if you're not sure you really want to play him/her.
However, if you know you're gonna play Kayle a lot, and really want to master her, these are the runes you will need:

Marks

Marks specialize in Physical Attack and are the best runes if you want Attack Speed - apart from Quintessences of course. Righteous Fury - as already mentioned - hurts A LOT when Attack Speed and Ability Power are combined. Therefore Greater Mark of Attack Speed should be chosen.
Alternatively, you could use Greater Mark of Magic Penetration. Each hit with Righteous Fury will hit harder, but you won't strike as fast.

Seals

Seals specialize in defense and Armor is a good choice. Greater Seal of Armor does not grant you either of this build's primary stats, but protection will definitely come in handy as it makes you less fragile.

Glyphs

Glyphs specialize in abilities. Choose Greater Glyph of Ability Power. This will make your Righteous Fury hurt a lot during the early-game ganks.

Quintessences

Quintessences are the best runes of all and you can only have 3 of them. They provide very high bonuses (for runes, that is). Runes are effective during the early game, but not very much later on. Greater Quintessence of Ability Power should be chosen. 4.95*3 Ability Power may not be very much late-game, but it makes a difference in the early fights. An alternative could be Greater Quintessence of Attack Speed which gives Attack Speed bonus.

One thing to know about runes! The tier 1 and 2 runes are NOT very effective. I would recommend saving the Influence Points and waiting until level 20 before buying any runes. You'll do fine without runes the first 19 levels.

Summary

This guide is pretty much based on Righteous Fury, so it is of course to be maxed early. At level 7 I upgrade Reckoning (which will also hurt quite a lot later in the game) and at level 8 I upgrade Divine Blessing. From there, I max Righteous Fury. The only exception is when I upgrade Intervention at level 6, 11 and 16. Reckoning should be maxed before Divine Blessing.
This is what I do. I like getting lots of minions during the early game. That's why I max Righteous Fury first. But if you'd rather deal some heavy damage to enemy champion right from the start, Reckoning should be maxed before Righteous Fury. Just keep in mind that Righteous Fury WILL deal more damage per second when you've got some attack speed.
Reckoning scales really great with Ability Power. If you follow this guide, it will deal 260 + 335 magic damage at rank 5. That's a lot in one hit. The base cooldown of the ability is 8 seconds, but it should be around 6 if you buy the items I suggest. It can spammed to deal some HEAVY damage late-game. Thanks to Kurimen for reminding me how powerful this ability is.

Summary

For minion kills

For bursting champions early-game

Farming should be easy! There really isn't much need for jungling. Maybe kill an Ancient Golem from time to time to get the Mana Regeneration and Cooldown Reduction buff. It will increase your sustainability because you can cast Divine Blessing more often without burning all your Mana, meaning you can heal yourself without having to worry about running out of Mana too much. Baron's buff is alway good, but don't try taking him down on your own!
Remember that the buff from the Lizard Elder is another way to get an on-hit slow effect!

The reason jungling isn't really necessary is that you'll get LOTS of minion kills. Righteous Fury is great when killing minions. No matter how many there are, they'll all be DEAD after a few swings with that flame sword! Later in the game, you don't even have to think in order to last hit the minions. It just happens.

There IS a tip I'd like to share, though!
When attacking a minion wave, try changing your target when the minion you were focusing on is really low. The AoE damage from Righteous Fury should take it down. This saves some time when farming. During the early game, you may be missing out on some minion kills, if you try this, but can't time your attacks correctly. But fear not! The S-key is your savior!
I don't think everyone knows about this trick, but pressing "S" will make your champion stop. It won't move, it won't attack. If you're worried about missing minion kills, hold down the S-key and release it, when you know you'll get the last hit. You're welcome.

Painful but "glassy"

Rushing for Attack Speed early in the game increases your damage per second quite a lot. However, you may be a little bit fragile. Don't get too close to those close-combat champions. With Righteous Fury you can damage them while still maintaining the distance.

Chasing

Reckoning hits hard once you get some Ability Power, but it also slows your target. This is especially useful when chasing an enemy. You can also use Divine Blessing as it does not only heal you. It also increases your (or an ally's!!) movement speed for a few seconds.

Intervention

At level 6 you are a pain in the butt to the enemy team. Be quick with that Intervention! It can really turn a gank around - or prevent an enemy from getting a kill. Imagine how easy it is to survive Karthus' Requiem with this skill! One time, an enemy Darius just couldn't figure out how I kept surviving his Noxian Guillotine. I told him I was a hacker. It wasn't true, but I had a lot of fun ( :P ).
Sometimes it can be difficult to target the right champion with Intervention if they're too close to each other. I have died several times because I misclicked and shielded another champion when I was the one, who was about to die. In these situation you can use this little trick I discovered (you probably already know about this. I just didn't and I was surprised). The small image of your champion in the lower left conner of the screen can be clicked to target your champion with an ability. That means that you can ALWAYS be sure to target yourself, if you are low on health. Click Intervention, click the image and you're INVINCIBLE... for a few seconds. EDIT: I found out that you can also hold down the Alt key while casting the spell to cast it on yourself. Thanks to Mizzle Sticks for reminding me.

Risky Kills

Don't you just hate it when you ALMOST kill an enemy... and suddenly he flashes back to the Turret? I do. But with Intervention you don't have to be scared of a Turret! Yes, it hits pretty hard, but you are completely immune to ALL damage for Intervention's duration. It's only 2-3 seconds, but trust me, it'll do. If you want to be absolutely sure that you won't get hurt, Ghost or Divine Blessing can be used to get out of the Turret's range in time.

Thank you for reading my guide. You should now be ready to smite some foes! This is the first guide I've published, so I'd like some feedback. If you have any questions I'll try to answer them. If you have anything to add, I'll think about it and maybe add it to the guide. I'll be open to criticism.
I wrote a guide to Shyvana once, but I never published it, because I wasn't entirely happy with it in the end. I may have been a bit lucky those times when it actually worked for me. But I AM happy with this guide, and I can promise you that you WILL be feared on the battlefield if you manage to get the items. Be sure to get some minion kills - and preferably first blood!

I will make some corrections if I've made mistakes, and I will add content to this guide, if I come across anything of relevance.

Enjoy! :D