I'd agree that jungle and mid are the easiest for climbing. As far as who you want to play it really depends on your play style. You generally either want someone who can snowball the early game or someone who can win team fights easily. Right now Graves, Hecarim, Vi and Kha'Zix seem to be the easiest in jungle. Champions with great engage/cc can also be extremely useful like Amumu and Zac. In mid lane it pays to play easy champs with tons of burst. Annie is the champ who immediately comes to mind because her CSing is super easy and her stun is point and click. With mid a really easy tactic is to play someone with a lot of wave clear like Ahri or Talon, push your opponent under tower, and then roam to another lane for another kill. It's really easy to tilt the enemy team by doing this repeatedly.
In season 5 I climbed out of bronze mainly playing engage junglers like Malphite, Sejuani and Amumu (yes I know Malphite is crap in the jungle but you can make pretty much anything work in bronze.) In season 6 I climbed out of bronze as ADC primarily using Miss Fortune (and Lucian and Tristana to a lesser extent.) I would recommend ADC in season 7 though.
Just keep it simple. The simpler the champion the easier it is for you to focus on learning other aspects of the game which is ultimately what's going to help you climb.