We all know champions are limited by the League when they are on the Fields of Justice, meaning that they can do a lot more than what we are actually seeing.

Having that in mind... What's the true potential of Karthus' Requiem? I mean, it can already hit all champions on the map. What happens when he's outside the League...?

Also keep in mind, he's only singing 3 seconds of it. What happens if he sings the entire song?

It's implied sorta that everyone would die
Why he doesn't do it.... I don't know. Karthus is mysterious like that.

There's a fan theory about him, saying that he's only singing one word of the song when he ults, and if he were to sing the whole thing, all but the most powerful beings in Valoran would die, but it would drain Karthus of all his energy. The only reason he feels safe using it on the fields of justice is because he won't use up any of his energy, and the League has protections around the champions there.

Well he definitely "Seeks to break his dark requiem upon the world". Why he doesn't do it at any given time? I'm sure it has something to do with the undead preventing him from doing it at the moment.

this is based on the old lore
there was a theory that karthus does not know the entire song but had it written down on a scroll that he lost to the mage who gave ryze his powers and then ryze took the scroll when he left for the league. this also explains why karthus entered the league, to get his scroll back.

Karthus is absorbing the energies on the field of justice until he is strong enough to sing the song without draining him and leaving him vulnerable, for that he needs ryze's scroll that enables the ultimate SPELL VAMP imagine his Q and his aoe skill maddd spell vamp!!!

He would probably implode before he finished it, same thing that happens in a lot of fantasies if a mage uses more power than they have

Here is another possibility. Karthus is a creature spreading the gift of undeath throughout the world. If he had a spell that insta-killed everyone, everywhere, at once, he could sing the whole song and obliterate everyone at once. Thus beginning the work of creating 6 billion undead.

6 billion/everyone in the entire world may sound like a great deal of undeath, but remember, Karthus is effectively immortal. Everyone in the world may not be enough. Undeath can only come after death, and death can only come after life. So his fighting within the league (an organization designed to stop wars) may be his way of preserving enough life so he can make even more undead.

To an immortal, they may plan centuries or even millenia in advanced. Karthus might have the power, simply not elect to use it.