Singed
The oldest League of Legends champion, whom Riot Games designed first, has a simple rule tied to him. Never chase a Singed! It's a rule that every League player knows, but not everyone follows.
He outplays his enemies by proxy farming, and placing himself in a situation where the enemy top laner and jugler think they can easily catch him. Only to lure them into chasing him, and whittling them down with his poison for an inevitable double kill.
Darius
One of the best LoL champions for kills, Darius, takes a different approach when having to 1v2 his enemies. With his ghost active for the added movement speed, he can position himself in the perfect spot for his Q to hit both champions, stack his passive, and help him heal through their damage.
From there, it's only a matter of focusing on one enemy to quickly execute them with his ultimate, and then transferring the stacks to the other champion, and using the Noxian Guillotine on them as well!
Illaoi
There is a simple reason Illaoi is one of the most dominant champions in League of Legends, and it's the same one that allows her to "outplay" her enemies in a 1v2 fight.
The reason is her ultimate, and it can't really be called an outplay as much as it is Illaoi stat-checking two enemies on her own, and winning every time!
The ability is even more devastating in a team fight, where escape is not as easy as it is when facing just Illaoi, so she can even be considered a 1v5 champion in the right conditions!
Tryndamere
The one thing all damage dealers struggle with in League of Legends is surviving the onslaught of damage directed their way, especially when it comes from two champions rather than one.
That is where Tryndamere's ultimate ability comes in, which allows him to defy death for 5 seconds, and continue dishing out his own damage.
Coupled with his incredible crits on his auto attacks, a lot of attack speed, and one of the best dash abilities in LoL that makes it hard to kite him, a good Tryndamere can burst down his enemies in a 1v2, before they are able to do the same to him!
Heimerdinger
A 1v2 fight in League of Legends is usually a gank against a solo lane. For Heimerdinger, it's his preferred battlefield, due to the very thing that makes him one of the most annoying champions in LoL.
He can prepare his lane with turrets that deal incredible magic damage on their own and block enemy skill shots. All Heimerdinger needs to do is position well enough to bait the enemy into his turrets, and not let them kill him by using his CC to stop them if they get too close!
Mordekaiser
While it may look like it due to his heavy armor, Mordekaiser is one of the best duelists in League of Legends. However, his ultimate, which makes him such a powerful duelist is something that works even better when fighting 1v2.
By casting on an enemy, he steals a part of their stats and separates them from their allies to fight Mordekaiser 1v1. Upon killing them, Mordekaiser leaves his realm, and can then take on the other enemy, empowered with the extra stats!
The only drawback of this ability is not knowing what is happening outside, as you may find yourself leaving your realm only to find the entire enemy team around you!
Katarina
If you want to play League and climb ranks quickly, it's widely accepted that the best way to do so is to master Katarina. She is one of the most popular champions in LoL for a reason, after all.
Her kit allows her to avoid enemy damage, with good use of her daggers and her Shunpo ability, while dealing damage in the process. Upon scoring a takedown, all of her abilities have their cooldowns either shortened or reset, allowing her to do the same thing over and over again.
In essence, Katarina is not just a 1v2 champion, but one who can take on the entire team in team fights and win!
Yasuo
Despite his reputation as an inting champion, Yasuo is easily one of the best 1v2 champions in LoL. Rather, he's the best one as he thrives in fights that involve multiple enemies.
His dashes only work when used on an enemy unit, so he can use enemy champions against themselves, by dashing through them and causing them to be in each other's way.
And, should the enemy jungler ganking Yasuo be someone like Sejuani, who can CC him with her ultimate, his wind wall will neutralize all skill shots coming his way. With these two abilities and his incredible damage, Yasuo is the champion to play if you want to win a 1v2 fight consistently if played correctly!