There is no "best 1v1 champ." While a DPS can walk up to you and instagib, a well-built Shen who stacked armor could counter that easily.
If you built Last Whisper to counter tanks, someone who didn't build Last Whisper could probably kill you from sheer damage.
Some champions are designed to function well in 1v1 situations. Of these there are:
Udyr, Jax, Olaf, any AD carry
Essentially any DPS or sustained mage has a chance of winning a 1v1.
If you take the time to start comparing skillsets, I would say Tristana is one of the strongest in a 1v1 situation. Highest sustained DPS endgame (similar to Kog), if you get too close she ults/jumps away, and the healing reduction means any lifesteal is halved.
But even she can be countered with any bursty mage if it's glass cannon Trist.