1. Thanks to dark25granti I have changed Wriggle Lantern into Hextech Gunblade, giving Kayle lifesteal, spell vamp, additional damage and ability power.
2. Thanks to Mastajdog, I have changed Greater Glyph of Furor into Greater Glyph of Shielding.
3. Changed Masteries because of the patch
4. Rewrote Masteries chapter
5. Added Pros/ Cons chapter
6. Added Situational/ Other Items chapter
7. Changed my runes, build and masteries completely and the chapters as well

I recently completely remade this guide and it is as good as new. I picked up Kayle again after getting bored with Yorick and I have to say, Kayle is downright amazing. Her potential as a carry is insane and this build will make her ******** dangerous. Anything that you don't like, or anything that you have a problem with just comment below and tell me what's up, I will change it... hopefully. Anyway, on with the guide and remember to vote after you've read it.

Pros:

With your E you become ranged!!!

With E on, you can benefit from both Lifesteal and Spell Vamp

Can chase people really easily with Q and W

Ult allows you to turret dive really well

On-hit damage like a boss

Cons:

Screwed when you're out of mana and a jungler comes to gank you

Squishy with this build

If you're CCed, you will be burst down in two seconds

Okay, first of all, Items.

My starting items are Boots, 2 Health Potions and a Mana Potion. The Boots helps me get to the brush near the enemies side on the top lane first and keeps me in range early to keep hitting him with my E. The potions do not need to be explained.

The items I rush next are Wit's End or Malady, Berserker's Greaves and Nashor's Tooth because:

Wit's End is for mainly for the AS/AP Hybrid stats and the Magic Resist. If the enemy AP carry sucks, or isn't ganking you, you can get Malady first for its more offensive stats. Also, it creates great synergy with your passive, allowing you to hit, all the more harder.

Berserker's Greaves gives you speed and attack speed which is the key thing to this build as you will be building up an exceptional amount of attack speed items later on. The extra speed goes well with your Diving Blessing and Reckoning ensuring every chase successful.

Nashor's Tooth gives you a lot of Attack Speed, Ability Power and Cooldown Reduction. The Cooldown Reduction is great and with your masteries, you can keep your E up all the time. The Attack Speed benefits your E and the Ability Power greatly impacts how hard your Q hits. The Mana Regen is great for Kayle because when I use Kayle, I always manage to run out of mana, so the Mana Regen helps.

After getting your first three items, you can then focus on the items that turns you from manageable to nightmarish. Hextech Revolver gives you ability power and spell vamp and builds up to Hextech Gunblade. Hextech Gunblade gives you exceptional spell vamp, exceptional damage, exceptional ability power and a really good active ability. Malady if you haven't already got it, benefits are stated above. Madred's Bloodrazor gives you even more attack speed, attack damage, armor and that awesome Unique Passive to shred tanks.

It has been brought up that there are some items that are good as well so I would put them into here, please comment to add more:
Black Cleaver is great as well and you should buy it when the opposition is investing heavily into Armor.
Guinsoo's Rageblade can be used to replace Malady or Wit's End late game if you want to deal more damage
Trinity Force is great for Kayle because she can use every stat from this item
Boots of Swiftness is great for Kayle as you need to constantly keep the person in range of your E
Frozen Mallet's passive is kinda wasted on Kayle. With Hextech Gunblade, your Q and E, It is really hard to get away from Kayle. Of course, the health helps and I love building this item when trolling around, shouting "I AM ASHE WITH A FLAMING SWORD AND WINGS!!!"

The Marks of Alacrity are great for early game and late game, synergizes with your E, letting you hit the opponent more times and shredding their magic resist, while buffing your own magic resist with Malady and Wit's End, quicker.
The Glyphs of Insight is for letting your E and Q hurt more. It goes well with Malady and your passive.
The Seals of Defense is because personally, as Kayle, I never was really scared about AP carries. I think a Wit's End is enough and my ult usually renders them useless against me and I kill them in a flash. But, after I kill the AP carry, there will be a lot of enemies left that deal physical damage, so I think it is more important for team fight survival.
The Quints of Replenishment is because I usually run out of mana as Kayle, and these Quints helps me gain my mana back quicker early game.

I put most of the points into offense (no surprise there) for maximum damage output. I need all the attack speed I can get thus the points into Alacrity. Invested points into ability power because it benefits your only nuke, Reckoning. The rest don't need to be explained.
The points I put into Utility is very useful as the extra mana regen helps and the movement speed allows you to catch up to opponents to slow them with Hextech Gunblade or your Q. The extra mana is great for early game and late game.

It's kind of hard to set a skill sequence for Kayle. I mean, you have to set your skills according to what you're up against. Personally, I max Righteous Fury first for maximum damage output, but I break this rule sometimes when I am up against a champion who I know could take me down, which is when I put a few more points into Reckoning to constantly harass them, denying them experience until I can take them down. Therefore, I conclude that you have to set your skills according to how your match is going.

This is how I use my skills.
I usually use Righteous Fury for farming but I also use it during champion fights for maximum damage output. I initiate with Reckoning obviously because of the damage amplification. I generally use Divine Blessing when needing to chase with opponents or if I really am severely injured (lovely for heal baiting). I use Intervention if the opponent is trying to deal a lot of damage to me in a short period, for example, when I encounter mages or when someone like Master Yi uses his ultimate and Wuju style. I also use Intervention to successfully dive turrets or bait them into turrets.

I take Ignite because with my Q, it deals increased damage and is great for stopping them from healing for a high amount of health when they're nearly dead. I take Teleport because I don't particularly like walking up to my lane when it's being dominated because then my opponent gets a lot more XP and becomes generally stronger than me. It is also great for flashing to another lane if your ally there needs help.

Early Game
Go top lane and get into the brush near the enemy's side. Wait till the enemy top laner walks really close to the brush, then click E and hit him continuously. This should bring him down to pretty low health and make him waste a health pot. In lane try to hit him with Q, self cast W and E him down to hell. This is crazy harass and should be done to zone him out, denying him XP and Gold if you can pull it off. If you aren't harassing, try to farm up as much as you can because you need the Gold. Ward the river and be wary of ganks as that is how people get killed in lane.

Mid Game
You should have 3 items already and be building up for your 4th. If you haven't already pushed down the enemy top turret, you should push it down fast. Afterwards, you can either push the minions to their next turret then go look for a gank or just look for a gank after the first turret is pushed down. Your Q + W + Hextech Gunblade combo should stop anyone from escaping your ganks.

Late Game
You should have your full build or nearly full build. You should still be wary of enemies as you are really squishy and you can be easily burst down. After your initiator has initiated, go in, Q the Squishy, E and take him down. If people are focusing you, you should Ult. This way, with their carries dead, you pretty much won the teamfight and after this, you should go and get Baron or take down their Nexus, if you can.

All I can say is that, Kayle's carry potential is crazy. I once went against a fully built Nasus and fully built Ahri, killed both of them and came out with more health than I started off thanks to my Ult and Hextech Gunblade. The point I'm trying to make is that Kayle is an amazing carry and shouldn't be under-rated even after the huge rework/nerf.