Any melee assassin, like master yi, with lots of crit %, very fast attacks or high % of critical damage bonus and the most lifesteal possible.
It is quite hard to solo Baron Nashor alone due to Riots implications to make it difficult xD. Very few can do it now, though my bet would definitely be an attack speed lifesteal built Warwick, and or Aatrox. Possibly full build Master Yi, once you have a last whisper in your inventory of course.
I used to be able to do it quite easily on Tryndamere but since the recent changes to baron I haven't found out what build I need / what level I need to solo baron. Before I used to be able to solo baron on about 3 items at around 22ish minutes or so. I know a jungler like Nunu can easily solo baron if they get tanky enough.
Pretty much any person with a good attack speed or damage steroid who builds lifesteal and has some semblance of survivability can solo baron once they hit full build and have adequate items and tankiness to deal with the debuff and dish out enough damage.
A lot of champions can do it but they need to get into the later stages of the game. I've seen a lot of people that solo the Nashor with WW, Udyr and Aatrox.
Any champion can solo baron. As long as you have a way to heal up damage, you can solo baron. Most popular champion would be Master Yi because of his ability to increase attack speed with his ultimate.
Not only melee assassins, but also ADCs, like Vayne. Of course that Vayne must be completely fed in order to be able of doing this, but regardless of that... it's actually feasible with any champion, probably except supports, which are overfed.
I think I have done this using Warwick and Nocturne but not until I have fully bought some stuff. Mostly some stuff with crit and life steal would do.
