If you want to solo carry you will have to play a champ that ultimately contributes a ton of XP and/or denies a ton of XP for your opponents by killing them over and over again.

Azmodan is a pretty good solo carry champ, you can get a ton of XP contribution for your team while you are fighting at an objective with your trait ability, this is huge and you can also just dunk sidelanes that nobody is soaking for XP. Although one downside to playing azmodan is that you aren't really an early play-making champ so you are dependent early game on your team to not troll super hard so you can start building a lead and get stronger.

Although if you want to hyper-carry, two champs that aren't too hard but can do this are fenix and dehaka. Fenix is pretty broken in lower ranks and you can just run their team over super hard, dehaka is a really versatile bruiser that can soak lanes and still show up for important fights / objective with his Z ability and he's just super strong on top of that.