Ok lotta questions here but I'll give it a go!
#Builds
Depends on what you're going for. Attack speed/AP builds are usually favored because that also increases her healing and usually gets her some cooldown reduction for her ult as well, however Kayle is one of the few champions that can do alright with a hybrid damage build because of her shred passive). Personally I would suggest going Attack Speed/AP to start, it's all around easier imho.
Try going: 
Now do keep in mind that build recommendations are a guideline and straying from them to better fit your specific circumstances is a good idea. For example this build has no tenacity, so going 
instead against a cc-heavy team would probably be a good idea. Or if you're going against a Nasus or Malphite (Who can destroy your attack speed) you may consider trading the Wit's End and Nashor's for more AP Burst items like Void Staff and/or Lich Bane.
If you're feeling gutsy you can try a hybrid damage build. These are tougher because while you'll do more damage overall you're relying on your passive for mitigation shred, which takes a bit to ramp up. So this is more of a sustained damage build than anything.
Overall your biggest form of damage is your E and your passive so going attack speed will do well for you unless, again, there are situation reasons you can't (Nasus, Malphite).
Note: Runnans will only be applied to your E augmented autos and not your normal melee autos. After you have enough Cooldown reduction you should be able to have your E up all the time but be aware.
Lich Bane
Lich Bane will proc after every spell cast. Your E activation will proc it, but not concurrent auto attacks. Keep in mind as well Lich bane (and Sheen) have a 1.5 sec internal cooldown. Meaning the spellblade passive will not activate if you spam your spells faster than 1.5 secs each. It's to prevent spam mages like Karthus from going out of control with the item.
#Abilities
Solo-lane (Top/mid) you'll want to max E first for the reduced cooldown, and then max Q second, for the reduced cooldown and increased slow.
Support Kayle is not the best idea. It's not the worst choice, but many other dedicated supports outshine her as far as sustain or crowd control in laning. If you do end up doing that though what ability you max first depends on how you're doing. Taking a lot of harass/poke? Max W first, if they outrange you that much your E will only be needed to push them off. Otherwise max E first and Q second. Levels in your W only gets you more base healing/speed and increases the mana cost of the spell so you get more bang for your level with the decreased cooldown, more damage and more slow from the Q levels.
Hope that helped! Feel free to ask specifics anywhere
