Let's start from the masteries and the runes. They give 17.5 Cooldown reduction and +15% attack speed at lv1, even just this is quite broken, but let's go on, you will se how this is so much broken in the middle game.

You take ghost to attack easier while moving or obviously for the movement speed and you take teleport to get in your line faster or gank the other to secure a kill having the number advantage.

In the line phase you should sit farming and due to your utility's masteries, that give you a +5% of experience you should be able to reach lv6 before your opponent, that's mean that if you don't feed, they should never be able to kill you.

While farming use your E to get the 400 range, don't worry to lose one or two minions, just don't fall behind your line's opponent. In the mid game it will be impossible to miss a minion, if you haven't feeded your enemies and even the other lines are not lost.

After your second back you should have Stinger and Fiendish Codex that give +20% of cooldown reduction and plus your starting 17.50, it will be a total of 37.50, just to remeber the max cooldown reduction is 40% but 37.50 is just perfect for you, because it will give a 10s cooldown on your E, that's mean you will have a permanent E due to the fact that the skill's duration is 10s. Thanks to this if you remember to press E every 10s Kayle will be a ranged champion with one of the fastest clean. You will not have to warry about your mana due to the Doran's Rings and without missing even a Cs your line's enemy will fall behind and he will not even be able to kill you due to your ulty. while you will be able to poke him and teleport to other lines to farm kills and assists.

Please don't consider the 27.47% cooldown reduction written on the top by mobafire, it doesn't count the masteries cooldown reduction that is a +10%.

So right now your E is broken, that make you a full ranged champion, dealing magic damage on basic attacks, why not get more attack speed? Runaan's Hurricane is a really good item for attack speed and due to the fact that your E make you a ranged champion you will even get the passive from Runaan's Hurricane for every basic hit (Yes theoretically this item should not work on kayle, but with the range from her E, it does). This mean that every basic hit will have the damage from the E, the damage form Runaan's Hurricane and the damage from Nashor's Tooth, and remember that those are all ranged damage on area and not just on one target, so if you will get fat, you might 2vs1 with no problem.

Kayle can counter a lot of champions and doesn't have any real counter, but at the same time you should fear any champion that has good mobility, stuns or knock up.
Yeah probably kayle has too many champions that might counter her if right played, for example Yi that will counter her ulty with the his heal or annie with the stun+poke etc... but your job is to farm all day and help other lines when your tower is not in danger.

If your enemy will leave the line, because they can't kill you, just SS and punish them killing the tower. if the other lines will not back or ward the river after your SS, just consider if the enemy might get a kill and if so use your teleport to help that line.

The real team fights, should start in the middle game and you should have started your Warmog so don't worry to die if you are not focused. Don't burn instantly your ulty, but use it on the focused champion, try to don't use it if he is the support or a tank, you should keep it for the adc or for you. In teamfights the worst moment is when you ulty someone and they run away with your ulty. that's when you know that you can't win team fights and so the game if you don't have a carry, so in those games don't ulty them again, just ulty yourself and let them die.