I made this build to help you out a little. This is how I would probably build Kayle if I played her. Just remember, this is my OPINION on how she should be built, not gospel. If you find something works better, or my order in the build isn't working for you, that is fine. Trial and error are the best way to make a built that works for you.

Ok, so you are going to want to ignore runes until you are a level 20 player. You don't get the good ones until then, so it is a waste of IP otherwise. Summoner spells and masteries, however, are important.

I suggest as summoner spells you take Cleanse and Revive. Cleanse removes negative effects on you, which will save your life. This includes CC effects, ignite, etc. Just remember to use it when things get bad. I suggest revive for you right now since you are still learning. This just instantly returns you to life. Other options include Ghost and Heal. Ghost is a GTFO button, and Heal... heals you.

For now, I suggest you focus on a defensive layout for masteries. This means more survivability, which means you can keep hitting longer. Once you get to level 22, dropping points into offensive masteries will be good as well.

Now what this will do for you is pretty simple. The first 6 points are giving you resistance to most forms of damage right out of the box. This amount can not be relied on all game, but will keep you laning in the early phase of the game pretty well. You can, alternatively, take health regen after level seven and bypass the dodge. Realistically, the dodge is not going to give you huge numbers (2% dodge chance). So that is something to experiment with to find what you like. If you do go with health regen, that will let you pop a point into improved cleanse later on, which will be useful.

Hardened Skin is simply the only useful thing on its tier worth putting points in, so there is not much else to be said there. This mastery gives you damage reduction. Flat. That is huge, so just enjoy it.

Veterans Scars gives extra health. I hope that is pretty self explanatory. Now as I said before, if you went with health regen at the second tier, now you can drop a point into improved cleanse. Do it. It will reduce the cooldown of the summoner spell by 30 seconds, which can save your tail. The rest of the defensive tree should be self explanatory.

In offense, focusing on AP gives you just a bit more umph to those spells. You can decide between attack speed and cooldown reduction on the next tier. I would actually suggest you go with cooldown reduction. It leads to spell penetration for your last mastery point.

Now I am going to remind you that most of what I just discussed will not affect you for a while. The best thing to do with this part of my explanation is to understand my logic in going with things.

As far as items... Ok, Doran's Ring is good for a first item for you. Doran's shield is another good option. The ring gives you AP and Mana regen. The shield gives you armor and health regen. I would suggest the ring when you are not playing with Carl and myself, but it might be advantageous for you to have more armor and health regen.

From there, I would encourage you to build boots and work towards Nashor's Tooth. Your boots should depend heavily on the team you are fighting. If you are going against a team with a lot of CC (crowd control, such as slows, snares, taunts, stuns, etc), Mercury Treads are a must. Mercury treads give you magic resist and reduce the effects of CCs. Outside that, Boots of Lucidity or Berserker Greaves are a good bet. Both are good offensive options. Depends on your play style there.

After you finish your NT and boots, I would suggest a rageblade. This is just a great item. It gets more powerful the more you hit people, so getting into a fight starts to get nasty. Simple enough to build, and you usually seem to grab it anyway.

From there, Crystal Scepter is a huge item for Kayle. The AP and health are going to help you out for the rest of the game. More importantly, however, is the slow effect the item applies whenever you target someone with a spell. You already have a slow effect from your Q ability, so combined with this and the enemy will be cut to crawl speeds. In addition, you can speed yourself up with your W, so chasing becomes easy. This can also be used defensively to get away.

The hourglass gives you a few things. First is a major AP boost, and then armor. This item also has an ability you can activate that puts you in stasis for 2 seconds. For Kayle, use this to hold off death for a few seconds when things are going bad. It can change the game for you if someone is really pouring out burst damage.

Finally, I the Hextech Gunblade. This is a crazy little item that gives you life steal, spell vamp, damage, and AP. Spell vamp means a percentage of the damage you do to a target comes back as health to you. Life steal is the same for basic attacks. This can keep you alive longer, but I would hold off on this item until late game when things get really crazy. You have to sell your Doran's item at this point, but the benefit far outweighs the cost. So whatever.