Although Kayle is a champion that is known for her weak lane presence, Kayle's support influence is paradoxically high. What holds her back in lane phase is her lack of kill pressure, abysmal waveclear, and being fairly weak to all-ins, all of which are solved in the support role. Since supports don't need to cs, Kayle can spend all of her mana focusing on harass, which makes it just as hard to stay in lane against Kayle as any other "Support" champion (i.e, brand, zyra, lux, etc.)

Goals: In lane, focus on getting all your Spellthief's stacks spent at level 1.
The best way to get this to happen is to stay in their botside bush and take a level one AA->Q->AA. This makes the slow last long enough that you can both get the final AA and get out reliably.
If they come into lane from the river, try and get a Q through their minions, and AA once or twice depending on how strong they are at level 1 (for example, probably not a good idea vs a Morgana who leveled Q). This works better with glacial, but klepto also gets more stacks
If you are vs a blitz or thresh, wait for them to collect relic shield stacks/ancient coins, then q from your side's bush and immediately go in for AA(s) before hiding behind your minions.

(it's important to note that it isn't a big deal if you can't get your stacks, but it makes laning just a little bit harder as you have to be a lot more aggressive with Q's and always proccing them on CD)

Once you have your Spellthief's stacks, focus on harassing with q whenever the CD is up. Hitting a Q whenever Manaflow is up actually generates mana, so don't be afraid to throw some Q's that might not hit if you aren't hurting for Spellthief procs. Once you have e, you can actually match the tribute stack generation, so you shouldn't all-in like you did at level 1 anymore. Ideally, you want to hit the enemy laners through the minions to push, because otherwise killing the casters would cost 55 mana extra, and ADC's generally can't harass and waveclear at the same time.

How you play the midgame entirely depends on how well you did botlane:

If you have a lead, playing a really aggressive roamer can work surprisingly well. Remember, your Q has resistance shred and a slow, so don't be afraid to engage first. The only way you will die is either hard CC or an assassin jumping on you after you've used ult, so save your ult until they have already used their mobility on (hopefully) not you. Make sure you don't just afk in the river/mid; you still have a duty to pressure botlane just as much as you have a duty to help the jungler in a 1v1, so don't leave bot while it's being pushed into tower just to help a Volibear 1v1 a Yi when he could have just easily walked away.

If you're behind, you have very few options. Roaming from botlane at this point would mean leaving your ADC farming at tower, thus denying them from scaling. Ultimately, a smart botlane will capitalize on this and will roam and get objectives, plus be an overall pain until you have items and start scaling. If they do this, use your mana to permapush the lane and make it hard for them to leave for extended lengths of time, but don't try and follow unless you have good vision of them. If they misplay and simply keep trying to snowball the lane, however, you can use their underestimation to your advantage. Communicate to your ADC that you're going to stay in lane for as long as physically possible, and recall only after shoving it into their tower. You're aiming to get a cs lead, and even if the enemy is up 800 gold it's not likely that they're going to invest that in defenses, which makes your poke still just as strong and hard to fight as ever.

(You can still leave lane when behind if you have a mid laner who can carry, and similarly you can stay in lane when ahead if roaming wouldn't accomplish much. This is a general guide, not a "Here's how to play League", and as such this advice is only true most games).

Once you reach late game, depending on your build, your team will either have a great engaging support who also provides invulnerability, or another carry who can evaporate enemies and make insane plays in teamfights. The game should be won almost naturally, but to make maximum effectiveness, you should play to the builds strength

AP: For AP Kayle, you're going to be near the front unless there's enough hard CC that getting hit by one would get you killed before you can ult. That doesn't mean you should be in front of your tank, but being alongside them until it comes time to engage will guarantee that your team can followup

AS: For this build, you're positioning just like an ADC by lategame. You won't have great engage or escape, but you'll always have the "**** you" button to assassins, and if they try and run away you can easily follow up. Don't go diving into the enemy team, just hit who you can after shredding their resistances.

This guide isn't complete, but this should be more than enough to have fun playing Kayle support. Good luck on the rift!