Teleport: This should be used for split-pushing and still maintaining a presence in team fights. Make sure you have your ult ready for use when you tele port into a team fight. When you get your runaans you we will be able to push to enemy turrets incredibly fast so let your teammates know when this summoner spell is available.
Flash: Self explanatory
Ghost: Take this only if you want to build a phantom dancer and you know the enemy team lacks any hard cc.

Start: When starting dorans ring compared to dorans blade keep in mind dorans blade against a melee opponent is sacrificing alot of on hit damage for pushing potential.

Core items: Rush nashors tooth This is kayles bread and butter item. Without this item you wouldn't have any dps and your heals will be lackluster. Since this is one of the only ap items you will be getting, get it as soon as possible to confuse the enemy team into building mr against you early.
Runaans Hurricane You can either get this or Black cleaver as your second item. If you know you have pushing potential on your enemy laner without being the center of your enemy junglers attention early, I highly reccommend Runaans Hurricane, since your pushing potential is outstanding(The best in the entire game in my opinion). Black cleaver if you are apt to fly solo in lane for a longer laning phase.
Optional Items: This is where hybrid kayle gets very interesting to play. Experiment with different magic pen and armor pen items to see what fits your playstyle best. Void staff alongside the new black cleaver allows you to get to a total of 45% armor pen and 50% magic pen on a single target.
Infinity edge is a snowball item
Phantom dancer sacrifices damage for great help for your teammates in team fights because the more armor and mr you shred faster allows your other carries to deal massive damage to tanks.
Triforce This is a late game item only if you have been split pushing correctly and have massive amounts of farm and gold by 40 minutes.
Frozen Mallet This is a great choice if you are fighting against a mostly melee team.
Boots: Berserkers Greaves... always... No questions. Just do it. Nike.

Kayle will defeat and outfarm almost all melee champions. Unless they have a hard cc ability like nasus or sion. Nasus isnt hard cc but it is hard when combined with jungler ganks or a 500-1000 damage Q ability that has a lower cooldown than your Q.