Kayle is a very easy character to play, she has abilities that allow her to pretty much attack and be played in any way imaginable. If played in the manner described below, you won't find a champ that can keep up with her Farming or End Game abilities. That said she is a slower starter and it may be worthwhile to get a few runes for her before you judge her too harshly, or just play with someone who can tank for you.

Marks - Alacrity is a must, all 9. If you notice above, her AS is what will stack her passive.
Glyphs - Intellect here is pretty important. It allows her bonus magic damage from her Fury to destroy all. This is pretty much the only thing that helps her Fury early game.
Seals - Fortitude helps a lot, simply because she is only a 4 bar hero to start and you won't be buying health items for a while. Her health problems aren't as bad at the start if you use these. I suppose you could use /sec instead, but you need the most help at the beginning.
Quints - I deck out her AS, so Alacrity is a beast with these. You could sub for one or two of either fort or intellect here if you have problems early game and need some support in a particular area.

Focus on Attack, getting the skills that help her beginning game. If you want to deal with mana shortages this is where you do it. I don't mind being short mana at times because her Farming is so easy with Fury, and Fury is so cheap on the mana, that you end up shopping often.

I focus on AS here too, notice almost all of the items you get are from that category.

1. Berserker Greaves - Boost AS and Movement speed. I ALWAYS buy the dagger first.

2. Phantom Dancer - Boosts AS, gives you a good crit chance, and boosts your movement speed. You really need the movement speed, particularly early game with short health. Also helps with catching fleeing champs and escaping ganks early.

3. Trinity Force - With all the boosts you get from this it doesn't make sense not to get it here. You can buy it in whatever order you want depending on what you need first. I normally do Zeal, Phage and then Trinity as a whole. This instantly fixes your mana/health shortages, gives you way more damage per strike and you can actively to select it every two seconds to boost your next strike. When combined with Reckoning and Surge you can easy do over 2k damage in under a second, but more about that later.

For farming and champ killing I use the same sequence.
You really want EQEQ to start so that you can Farm with E and do way more damage from afar when combining E+Q. Combos discussed in their own section below.
Getting one W and one R will help survivability but after that lay off the W and R till the end. (Explained below)
More EQEQEQ, at lvl 12 you should have E and Q maxed.
Then max W.
Then lastly finish up your R's.

Q - Reckoning, Does a decent amount of damage but MORE IMPORTANTLY slows the target champ and you do MORE DAMAGE while they are slowed. When you can stack your passive on a slowed enemy and use your E to boost damage and range, you SERIOUSLY weaken them to the point where your attacks do a TON of damage and with AS maxing you do it VERY VERY FAST!

E - Fury, Increases your range, and adds bonus magic/splash damage to each attack. Lasts quite a while and is by far your most useful farming tool. Kills groups of minions very quickly due to it's splash and you want to max it early.

W - Divine Blessing, it's alright but not important early game. It's what makes her a support character, but you can use it on yourself. It boosts movement speed and AS most importantly. When getting ganked later in the game it can save you or whoever needs it most. Also important but mostly in late game when groups start to switch up.

R - Intervention, kinda sucks. Makes you invulnerable to all damage for a few seconds but doesn't last very long. Timing is key. I only get one early when I'm playing against champs with Crazy Ults, and only as a countermeasure to them. It helps save your Teamies if you use it then trigger your blessing on them too. It'll save you, but often only temporarily. Just sucks because you have to click on yourself after selecting it. Easy to accidentally click on the wrong person in team battles.

Passive - not sure what it's called, but it lowers the opponent's defense stats every strike. stacks only stay for a second or two but will stack up to 5 times. So getting AS items allows you to fully use your passive. This is the reason the can do more damage when boosting AS then when boosting AD. Hitting the max AS of 2.5 strikes/sec will allow you to keep the opponent fully stacked up and will kill them very very quickly.

Kayle has one of the best farming skills in the game. Fury makes your normal attacks ranged and have extra splash damage. That splash lets you wipe out huge crowds of minions very quickly. Farming with her is even easier/faster if you use surge.

Her E+Surge combo is also HIGHLY effective on turrets! :)

Surge is important for towering and just to help out with champs/minions.
Heal is replaceable but it certainly helps early game.

E first, then Q is your generic attack on an enemy champ.
Can be expanded to : E - Surge - Q , and later E - W on self - Q or E - W - Surge - Q for maximum damage and slowing. This combo is freakin great and often totally kills squishy champs in as little as 4-5 hits which is less then 2 seconds at lvl 18.
Tips: Timing is easy with Kayle but knowing when you have another Reckoning available is what'll make you unstoppable late in game.

* Once you have trinity use it's passive is based on casting so your EDW(Self)Q combo will deal insane ammounts of damage. With a full build as listed above you can easily deal 3-4k damage in two seconds when your cooldowns happen to line up. I've killed tanking Shen with 5k health in just under 4 seconds before.

W - on self. Yes this is a combo, by lvl 18, you can cast this constantly and never run out of mana. Plus the bonus will last longer than the recharge when maxed and doesn't fade until it ends.

R + W, will save anyone from a gank at lvl 18 provided they dont get stunned. works 9/10 times but don't focus on it until the groups start to change and ganks start happening.

Kayle has 3 distinct phases, Early phase starts the game(obviously) and goes until you have Phantom dancer.

Tips: Focus on the EQ combo for needling enemy champs and farm minions as much as possible.
If you have very aggressive opponents then hang back and use your E to catch the melee minions.

YOUR GOAL HERE IS to level up to level 3-4 so you can start doing real damage to enemy champs. Farm as many minions as you can and Q the enemies to keep them out of your way.

Buying the dagger first lets you contend with groups of minions fairly fast and you need the cash for this build.

Shop when you have enough to finish the Greaves, if you are facing hard opponents, or just wait and get cash for a few extra daggers if you can.

Once you get the boots you are a little safer, and can more aggressively needle opponents and using that EQ combo you might even be able to get some quick much needed cash from a kill or two. Kills are NOT your goal early though as they come in bulk later. Farming is your priority.

Middle game is from Phantom dancer to the point where you have trinity and are around lvl 12-13.

Keep farming those minions and push the turrets for extra cash. Kills are not as important as keeping the opponents back so needle them with Q until they are low then E+Q them to death but don't chase if you would be taking any risk of a gank. Keeping your distance by using E seriously helps you avoid ganks. Just stay strong, the good part is coming.

Around lvl 14, provided you aren't behind in lvls too much, you will start to see that E+Q combo begin to Rape. Keep buying your items and as soon as you have enough cash get ionic spark for it's passive. It rocks. Chain lightning + your Trinity + E + Surge + W on self + Q is such an amazing combo I can't stress it enough.

You get a little wiggle room with the last two items, just be sure to keep them in the AS category. I just prefer the ones listed. Try to see where the enemies are building towards and counter it with your items. If they get really high armor stats go for AP and LS items to compensate. You can have fun picking new items each time because the first 4 are the most important. If it's tight though, use bloodclaws/razor as the fifth item because it's passive is NUTS.