My IGN is Nabura and I play on the OCE server, as I am writing this guide I am 250LP Grandmaster and am aiming for Challenger. I've been a Kayle OTP for about 5 years and so I've played her before and after her rework with a cumulative total of around 1.8 Million points on her with probably close to 900ish ranked games on the new, reworked Kayle.

Kayle is a Mid-late/Late game hybrid hyper carry with DPS and supportive teamfight capabilities with flexible build choices that has the potential to split OR teamfight depending on the enemy, and her own team compositions.

As she currently stands in top lane, Kayle is looking to be a niche counter-pick to most tanks, with lots of difficult bruiser matchups that require high wave management knowledge with good spacing mechanics to outsmart and outplay. She will be more likely to be split pushing in this position as Kayle top can sometimes create teams with no front line, which will lead to splitting becoming the win condition.

In mid lane, she fulfills the same role, although you won't be taking up a potential tank/engage slot and team fights will likely become a viable option with front liners to soak abilities for you. Splitting is still a completely viable option when vsing enemies that are hard to match in teamfights. Think control mages like Orianna, Glacial Augment champions or high range 1v9 ADCs like Aphelios/Lulu, Caitlyn, Kog'maw.

nabu#8860 on discord. Feel free to ask me anything Kayle related or anything regarding general fundamentals.

Most of these synergies are "balls to the wall" sort of champions that thrive in the middle of the entire enemy team. Supports like Lulu can also be amazing given you have enough frontline to make up for an ardent sup.

Something I like to do is look at OP.GG for what rune shards the enemy laner took and taking long sword if they go MR and dorans ring for armor. An extra thing to note is that PTA's damage type changes depending on whether you have more AP or AD so use this to your advantage vs high armor/MR base stat champs.

Corrupting Pot is the most popular start but can make last hitting a little hard. Personally I never buy this item because I feel that Dorans Shield and Long Sword both offer better stats for the two types of circumstances that you will find yourself in during laning. One other thing is that Corrupting deals DoT, so it'll help stop passives like Garen or Malphite from resetting for an extra 3 seconds. It can also have an opposite negative effect for anyone that starts with a Doran's Shield, so keep that in mind.

Doran's Ring is my default start against 95% of champions. Makes last hitting pretty easy and gives mana regen for a few more spells in early laning.

Doran's blade/Long Sword is a SUPER aggro start that is good against anti AP champions like Sylas, Galio, Kassadin that you can actually punish if played correctly. If you are going for AD early you may as well buy long sword for an earlier vamp scepter.

Doran's Shield is probably your buy against Cassio, Zoe and most poke Aery/Comet champions. This is actually an item that I have been utilizing a lot recently since it can allow you to still go PTA while letting you still survive through the laning phase.

I'm going to give you the same answer that everyone else probably gives you, it depends. A general rule of thumb to follow is to split against long range mages and tanks and to group vs bruisers and assassins.
Split vs - Malphite, Ivern, Orianna, Twitch, Lulu.
Teamfight vs - Irelia, Khazix, Talon, Vayne, Pyke.
These examples are pretty extreme but you get the idea.

Tethering is a concept that can be applied to any champion against any poke spell in the game. The way to win in these matchups is to run the enemy OOM.

Pretending to walk up for cs to bait a Syndra Q and then dodging it is this concept in its simplest form.

Once you run a mage OOM, you have two options. Constantly shove waves under their tower and make them miss CS or freeze the wave to make them miss CS. If you know the enemy has used TP and has no fast way to reset their HP/Mana then freezing can be very strong as you either deny multple waves of XP and Gold or force the enemy jungler to come to your lane to break the freeze for his laner.

I learnt this mainly from LS but Coach Curtis recently put out his own video on the topic. Both are good watches but start with LS's 15 minute video