Kayle has great burst damage and is good at chasing down foes, and her ultimate is very helpful in both teamfights, 1v1's and saving allies. Remember her heal also provides a speed boost, so you can use that to help catch up to an enemy. When you get your Frozen Mallet, use your E to slow enemies so you can chase them better. Once you have BOTRK and the Hextech Gunblade, you can easily chase or kite by using both actives to steal move speed and reduce the enemy's move speed. Your ultimate prevents your target from taking any damage at all, including summoner spells like Ignite, which makes Kayle a lot more powerful when 1v1'ing since she can negate all incoming damage. When playing in lanes be careful of junglers and champions with both gap closers and CC - your W wont provide much help in those cases.

If you are playing against a champion like Pantheon or Xin Zhao that has good damage and a stun / knock up, it may be better to play safe early game and avoid fighting. When you reach level 6 you will be much stronger in 1v1s, but if you are still having trouble simply wait for your jungler to gank.

In teamfights you are crucial to your team with your ultimate, Intervention. If your team engages, for example, a Jarvan IV entering with E-Q and ulting the adc, it will be great for you to ult him as soon as he is in and taking damage so he can come out unscathed. If the enemy team engages, and your team cannot fight back or they die too quick and you still have your ultimate, use it on a carry (mid or adc, etc) to help them survive or even get a kill or two after exiting alive.

Kayle can also be played in the mid lane. Since Kayle has great burst with her E, you can easily engage onto champions like Karthus at level 2, when you get your Q to provide a stack of Holy Fervor. Be careful about spamming abilities since you will lose a lot of mana from Q and W, and they even gain mana cost in levelling. Your E stays at 45 mana cost every level, however, so you are able to use your E more, which is your main source of damage. Remember to save mana for your ultimate, Intervention, when you are fighting.

Do not use your Q to kill creeps as it will drain your mana. Your E, though, is perfect for clearing large groups of minions quickly, since it has splash damage.

Pros:
Great early damage
Good survivability with heal
Decent at kiting

Cons:
Weak to gap closers
Will do bad if low cs and a lot of deaths