Flash
Is a really good summoner skill because you can use it in so many situations like escaping ganks , closing caps and go for surprising Flash+ Reckoning combos.

Ignite
Is the best offensive summoner for Kayle to focus down your targets down more quickly and pressure them in lane. The damage gets increased by Reckoning.

For masteries I recommend 21/9/0 with huge damage boosts to harass more effective early with your range advantage and to shut down enemy carries even faster.
The 9 points on defense are for an even more easy lane phase and for the most part to reduce minion damage while harassing with your Righteous Fury and getting some extra Health for a more late fame survivabiltiy.

Greater Mark of Attack Damage
Is the best way to get damage in lane to a harass/dominate and even makes last hitting easier at tower if you get pressured. Is a lot better than Greater Mark of Desolation because it gives you more early damage and scales with your skills.

Greater Seal of Armor
Is a nice source of armor and helps you win trades with enemy bruisers. Also reduces damage taken by minions while harassing.

Greater Glyph of Scaling Magic Resist
Is really important for not getting one shotted by enemy AP-Carry late game before having even an impact in a fight.

Greater Quintessence of Attack Damage
For same reason as marks.

Start:

Boots + 3 Health Potion
Is the best way to go because you can sustain after trades , escape ganks easier , dodge skills easier and chase down enemies with Righteous Fury.


Early:

Doran's Blade
Is a cheap early item with really useful stats no matter you are playing against for dominating lane. You should get at least two to get some early power , or three if you are losing lane to catch up faster.


Core:

Mercury's Treads
Should be build on Kayle for safer teamfighting in terms of CC reduction and less danger of getting killed by enemy AP-Carry.

Wriggle's Lantern
For lane dominance , lifesteal/sustain and in general good stats for trading with bruisers. If playing against an AP top lane you should just build Vampiric Scepter and get real damage items more quickly.

Phantom Dancer vs Infinity Edge:
Both can be build as first big damage item , but i generall build Phantomdancer first because the attackspeed just stack so well with Righteous Fury and helps chasing down targets with the movementspeed.
If you just want to take one target down really fast you can buy Infinity Edge+ Berserker's Greaves but you'll lose sustained damage and utility.


Teamfight Items:

Infinity Edge
Asap to get the most powerful damage item to support the Phantom Dancer and start killing Carries before they even realize.
The critical strikes in combination with Reckoning are so scary for enemy team.

Last Whisper
Should be bought next to be able to fight enemy bruisers late game and make your damage insane in combiantion with your Holy Fervor. Because your team has usually two AD-Carries with Kayle enemies will stack armor what makes this item even more important.

Quicksilver Sash
To prevent dying in teamfights after a bruiser or AP-Carry got any kind of CC onto you and also helps reducing taken magic damage. At all most cost efficient defence item for Kayle.

Bloodthirster
If the game goes like 45-50 minutes+ you can replace Wriggle's Lantern to get even more damage and insane teamfight sustain.


Situational:

Ninja Tabi
Against heavy bruisers early game like Lee Sin/Renekton/Riven/Darius for not being dominated and dealing with their early pressure.

Berserker's Greaves
If you dominate your lane and want to go with the Infinity Edge first path to get the needed attackspeed.

Frozen Mallet
Can be purchased rarely if you are winning your lane really hard and you want to snowball the other lanes too. Actually not needed because you already have a movementspeed buff in your Divine Blessing and a slow in Reckoning to chase people.

Guardian Angel
Can replace Quicksilver Sash if the enemy has no scary CC to focus you down and you are really important for your team. If you combine the item with 4 rawr damage items turns you are at a point where you dominate every teamfight and can take out 2-3 enemies without dying.

Intervention
Should be maxed everytime you can because it's so powerful and gives you the needed time to shut down other carries.
You can put this on yourself or the ad carry in a team fight to get as much time to deal damage.

Reckoning
Is the best basic skill to max because you can harass enemies easy , slow them down and increase your damage dealt on this target for a huge amount.

Righteous Fury
Is the skill to max after Reckoning because it gives a nice damage AoE splash that scales very well with you early Phantom Dancer.
The range advantage makes you so strong in top lane to harass bruisers and chase them.

Divine Blessing
Should be maxed last because you only use this for the movementspeed buff to escape and chase so one point is enough.
Don't spam it for the heal because you'll run out of mana and have enough sustain with Vampiric Scepter.

Winning:

In General Kayle is strong against enemies that can be kited like Darius, Riven Warwick and Udyr because this way she can win every trade and harass them really easy.
If they run at you just Reckoning on them and attack them with Righteous Fury until they reach you. You can outrun them with Divine Blessing if they want to chase you after Reckoning.
If Kayle can dominate the lane she will outscale everyone of those top laners late game.


Losing:

Kayle loses against heavy ranges harass champion that can pressure her more than she can like Yorick , Rumble and some bruiser that can go onto you , win trades and go out before you can counter harass them as hard as they did like Jax , Renekton.

In General Kayle can also be countered by stacking armor and building Frozen Heart / Randuin's Omen.
Before Kayle build a Quicksilver Sash she has also a really hard time against hard kinds of CC's but later she can just use this item and activate Intervention for free damage.