I play the balls out of Kayle right now and I build as a supportive tank. I have a very specific early-mid game build that I have tested and tweaked over about 20+ games as Kayle since the recent buff patch.
Skills - Max Q and W first grabbing a single level of E (ranged) at around 4-8 for that extra ranged power. Ult at 6, 11, 16.
Here is my very tested, very thought over, theory-crafted item build in order.
1: Mana Manipulator - because you can heal yourself and this gives great synergy with a mate
2: Meki Pendant - again, you can heal yourself so this really ups your harassment
3: Boots - obvious
4: Farie Charm - more mana regen - because spamming Q takes a ton of mana.
5: Amp. Tome - to make....
6: Fiendish Codex - -CD, Mana regen and the early boost to AP makes reckoning a monster (creep kills, harass, killer - 300+ dmg nuke on a 6 sec CD)
8: Cloth Armor
9: Heart of Gold - You are not a farmer so having a nice HP item that gives 5gp10 is awesome. I very rarely end up selling it with my item build
10: Boots of choice - Tabi for DPSers, Mercury for magic and snare - Berserkers works too. No need for more speed - your Blessing will give you 420+ speed with lvl 2 boots.
10: Giant belt - to make...
11: Soul Shroud - Great aura for you and team - -CD is amazing on Kayle (27 sec Intervention CD!) and its a great tank item (3000ish HP at this point)
12: Finish Stinger - that is why you bought F. Charm and also to make....
13: Nashtor's tooth - now you COULD make this before Soul Shroud, but I like to be more tanky and Nashtor doesn't give a ton of stats for combining (you could keep Stinger and Codex separate if you want for most of the game)
Also - you are now at max CD! 4.68 sec Reckoning (4 sec duration), 15 sec Fury (10 sec duration) and 27 sec CD on intervention at 16.... oh yeah... hardcore support
Now things get more situational
Personal favorite - Aegis of the Legion - because you are supportive in nature this works really well - it also makes your heal better than simply HP stacking. The dmg is just a treat.
Rylai's scepter - More HP than AoL and wayyyy more dmg than anything else but the slow is kinda "meh" as you only get to use it with Reckoning. The AP is a monster on your heal, nuke and on your Fury dmg. It is also more expensive than Aegis.
Warmog's - Welcome to 4000+ Hp - you are pretty much unkillable with any of these items, but this one makes you a juggernaut - problem is that it does nothing else and doesnt support. But boy can you soak dmg.
Other items to consider - Sheen (because you spam spells), Malady or Wit's End (to improve your Fury attack speed and dmg. Phage make you ever tankier and is rather useful. Atma's is also good but expensive - better with Rylai and Warmog - very hard to get all that done.
Most games end before I can get the final item unless we did Baron Nashtor or kept up on killing Dragon. As I said - you are not a farmer so money will be a bit of a problem.