I don't think there's any 1 champ who's best for 1v1 scenarios, but I have gotten into quite a few of them, typically with rampaging Yis, and come out on top with:
Mundo: He is so hard to kill 1v1 for most people and can rip the health right out of you with that cleaver.
Gangplank: A lot of people like to go toe to toe with me when I play gangplank and I don't get why, I guess because he has that sweet gun they forget he has a poisoned anti-heal sword, absurdly good crit chance and moves and attacks faster than anybody else.
Heimerdinger: While I haven't had much experience netting 1v1 kills, I've been jumped 2v1 and killed both of the enemies. It really depends on the state of his gun nest, though, but very rarely do I die as Heim without it being an organised "kill the engineer" attempt by the enemy team
in my personal opinion the "best" 1v1 champion is probably a fully feasted cho'gath. or a RIDICULOUSLY farmed up sion with like, two million HP or something stupid.