This is a guide focused on winning as Kayle.
Not dealing the most damage, not getting the best score, not getting Quadra Kills or Penta Kills, just winning.
While doing that you might happen to get fed too, but the main goal is winning.
Kayle is a very good champion for winning when you're thinking about strategy, she's quick at taking objectives, she's very easy to farm with against any match-up, she scales very well and is very, very good at team fights.
As Kayle you could be supportive when someone else is doing well, and you could hyper carry when needed.
She's also not too complicated mechanically, which gives you more room to think about strategy.

During Laning you usually want to play defensively, since you scale well into the late game. Kayle is very good at farming safely, and at escaping ganks.
You don't really depend on cooldowns with this build since you can consistently deal damage with your E almost all the time (15% CDR at level 1), thus allowing you to heavily win trades once the opponent uses their abilities to clear or try to poke you.
After level 6 Kayle can roam efficiently as she can push her lane easily and dive turrets easily using her ultimate. Having AoE sustain damage also helps winning 2v2s and 3v3s.
When you feel stronger than your opponent in an all out fight, you can farm without your E, and then activate your E only when they get too close, this is useful so that you can freeze lane and deny enemy minions, and works best against melee champions.

Lane note:
Don't heal yourself when you're low on Mana, and don't drink potions if you have the mana to slowly heal yourself to full.

Overall I wouldn't recommend buying many potions as Kayle, I usually use 5 or less potions throughout the whole game, and if you're planning on using more than that you might as well get a refillable.

As Kayle I would always recommend only doing things when you know why you're doing them.
Don't try to do a lot of pointless things at once, and you'll slowly find yourself becoming the mastermind and the shot-caller for your team.
Now, this doesn't mean play defensive and never do anything, you don't want to make running away your default action when you don't know if you can win a fight, what you want to do is try to think fast and analyze what's going on, here's an example:
You pushed your lane into the enemy tower, and are now heading towards bot lane. You see the enemy bot lane is pushed and you want to punish them. What you need to do now is, Look at the enemy bot lane's levels, your bot lane's levels, check if anyone's low on mana, read the chat to see if anyone pinged someone else's summoner spells, look at your allies' summoner spells, look at where the enemy mid laner is and try to figure out if the enemy jungler could be around based on where they were last seen, check if the enemy team has a teleport or a global ability like a Shen or Gangplank ult. Now, after you took all of that into account, you want to think about the risks you're taking, and the possible outcomes, and only then decide what you want to do.
This all may sound a bit too complex, and not practical, but you start doing these things unconsciously if you play with the right mindset.

When you force the enemy mid-laner to back, and you think you can outduel the enemy jungler, you push and damage their tower.

When you see the enemy team has 1 dead member, and 2 members in top lane, while you're 5 people near mid, you take dragon.

See a pattern here? Yup, you guessed it, don't let free objectives slip away.
This sounds basic but is nevertheless important.

You could take advantage of many things to get objectives, whether it's the enemy team being low, them being weaker than you and unable to win a 5v5, them having dead members, you having waves pushing towards them in side lanes, your team having a teleport while their team doesn't, etc.

Now this of course applies to League of Legends in general, but the reason I'm putting these things in my Kayle guide is because this guide is for winning, and Kayle is very good at winning through strategy and objectives.

NEVER, give up, as Kayle.
You scale better than most people think you do.
And you always want to force the enemy team to stay focused until the end, some people think you lost if the enemy team has a big advantage, I think the enemy team CAN win if they have a big advantage, but for that they need to slowly push their lead towards a win. You want to force them to do that. You want to only lose to very good players who take advantage of their lead and don't make mistakes, because mostly the enemies will still make mistakes when they're ahead, and you may be able to take advantage of those mistakes and turn the game around. And even if you're sure you lost, you don't type in chat "it's lost". You just do your best until your Nexus is destroyed.
Sorry for getting a bit off track there, that's not really Kayle related any more is it? Hehe.

Even if I think I might be able to win a 1v1, sometimes I won't take it.
Because I know if I don't get a kill now, my team will be able to win the upcoming teamfight, but if I die now, my team would not be able to fight.
Use the fact that you have insane team fights.