Kayle is a champion I have been playing extensively in unranked play with several friends for quite some time. I have tried a large number of builds with her, AD Heavy, AP Heavy, Full CDR, etc. This is a build I find works very well to her strengths. Courtesy of her passive, she gains AP from AD, so an AD Heavy build still grants her a large amount of AP. I also found that while CDR is nice, there really isn't a NEED for Righteous Fury to be up at all times.

Kayle can be a wrecking ball if you get a couple of early kills or assists, and she does deceptively high damage at low levels if you harass correctly.

For runes I go with AD Marks, Flat Mp5 Runes, and AP/lvl Glyphs and Quints. I find the flat AD runes give her a much needed increase in punch early game when she is weakest. Flat Mp5 runes because Kayle is a mana hungry ho, especially early game. I find that she has fewer problems in later game, but the items in this build aid in that. Ap/level Glyphs and Quints are standard for Kayle. They aid here also because this is an AS/AD heavier build.

I build Kayle with the defensive tree because she is very squishy despite her label as a bruiser. This helps her considerably especially early game. I run clarity for laning and group fight assist, so Improved Clarity is a must. Mana/health regen and Imp Ghost are a must and the extra experience gain is a huge boon.

I have heard mixed reviews on Manamune on most characters, and most people have said not to try it on Kayle. In my personal experiences, if you have any sense of playing her correctly, Manamune is almost a must. First, the AD scales extremely well, and the extra mana is a godsend for you early. Since you're constantly auto-attacking, she is constantly gaining mana, and it overall just aids significantly in the process. Boots I choose the 'Zerker Greaves because attack speed is Kayle's friend. From Boots I pick up a Stinger, but not the full Nashor's Tooth yet. At this point you build a Guinsoo's Rageblade, then finish off your Nashor's and put on a Stark's. Most people build Guinsoo's earlier, I know, but I find that it is a better synergizing weapon later when you have the attack speed to stack is much faster. Finally, if your game lasts long enough and you don't have situational items you want, I go with Lich Bane. Other options that I find work here well are Malady and Trinity Force.

I prioritize Intervention, then Righteous Fury, then Divine Blessing, then Reckoning.

Reckoning I find is more useful as a slow and increase to your main attacks, and as a nuke is fairly limited if you don't go full AP. The heal is nice for laning, though do not rely on it. It's really mana inefficient to constantly heal yourself up.

Ghost/Clarity

Ghost is your friend for chasing down champs and getting away with your ult. Clarity keeps you in the lane for longer. She's a character that needs a solid farm and needs to stay in lane as much as possible, so anything to help with that is good.

Laning -

Hang back and wait to last hit. If you see a melee champ going in for creeps, punish them with righteous fury and reckoning. Do not over pursue. Kayle is fragile and has no real hard escape tools.

Team Fights -

I find myself standing a bit back in team fights. You're a ranged and in this build you have limited range. Pick a target and go for it, but don't run into the teeth of the other team if you can avoid it. Your reckoning/righteous fury combo can put serious lead on target if you are given the opportunity, and in many cases your ult is a gamebreaker for teamfighting.