Get off and stay off
The way to kill enemies in Super Smash Bros. Ultimate is to simply get them off the stage, and keep them off the stage. Obviously. And the easiest way to do that is, um, hit them.
Once smacked off the stage, simply hit them again before they manage to land on the stage, or before they manage to grab the edge.
Jumping off stage and pushing your opponent back before they can recover is the most obvious way to do that, but of course that just means your opponent will make it harder for you to hit them. The first thing they'll do utilise their air dodge.
Projectiles, like Mario's fireball, are another great tool to distract off stage opponents.
Dealing with air dodge
The air dodge in Super Smash Bros. Ultimate has been buffed and nerfed in ways since Super Smash Bros. for Wii U. Here you get a directional momentum added to it, allowing you to reach for the ledge or avoid attacks more efficiently.
The downside to it though is that after an air dodge, your opponent is vulnerable for a decent length of time, even after they land. You can use this time to get in a hefty smash attack.
Once you know your opponent is air dodging reliably, wait for it. Jump at them, wait for an air dodge, and simply attack when their invincibility ends, or when they land.
If you're smashing enemies up or to the sides, then your opponent is likely to opt for a high recovery - returning to the stage from a high point.
This makes getting them off again a little bit easier - you'll be able to see where they're coming from clearly, and giving you decent set ups for a spike, if they're returning high from the side of the stage.
Predicting movement for spikes or simply batting the opponent away from the stage is the easiest way to deal with high recoveries.
These are trickier to get at, especially depending on the character you play, and who you're against.
But, for many characters, recovering from a low point can be risky. Though characters with a large Up Special will opt for it as it's often safer than recovering from high.
Though it has its own risks. When enemies are recovering low, you can drop off the stage and use a back air to slam opponents into the stage itself, hopefully bouncing them into the abyss. Though this can be tech'd by pressing shield as you hit the stage.
Another option when enemies are recovering low is to use a multi-hit move, such as Mario's down air or Pikachu's back air. Enemies coming up should collide with this, and hopefully again be plunged back down below.
Of course, before you should try edge guarding low or recovering low yourself, you should ensure you've mastered your character's recovery options, including jumps and special moves.