I am trying to get a smaller champ pool. So I am wondering, if you could only play one champ to work your way up the rank ladder who would it be? This is not your favorite champ, or the most fun. Who is the best to carry with and win games to get to the top?
The champion you're best at. There's not a "best" champion, because your playstyle might not mesh well with a champion that works great for other people. Even when azir was broken, I was terrible with him. Even when taliyah was "weak", I did well with her.
im not biased
Damage, Utility, Dueling, Team Fighting, she's the full package mate.
Best to carry? prob Hecarim since he is straight forward, simple, powerful, passively strong and various build pathes so you can be a tank or an assassin.
but i rock my baby Galio.
I've settled on Xerath. Enough wave clear to stall out losing games for a chance at a turn around, good burst, and an ultimate that's simply superb in lower ranks.
Your teamates started a 4v5 while you're half a map away? Good thing your level 16 ult covers half the map, you can contribute right from where you are.
Which lane?
Karma! it can be played in support mid or top!
Biased opinion is: 
Best for climbing? 
You may find me crazy, but 
.
Does a fuck ton of dmg and can help other lanes. Plus a stun every 6 sec.
Yasuo! 
easy? YI, Kat, Akali , Yauso
Faceroll and steamroller as hell if get going
Riot doesn't like you playing/maining only one champ (known as a one trick). Your desire to climb elo easier and master the capabilities of this one champ that can carry you, will be curtailed at every corner through the use of things like autofill and in some cases, your account being permanently suspended, should your one trick champion pick in any way, shape or form be contested by your teammates in solo queue. If it causes them to rage/grief/collapse in shock because you picked something outside of their understanding (i.e. soraka mid or teemo jungle), it will be you who is punished should enough players report you for their own grieving.
On a serious note, just play any one of these and get freelo:
Top lane; Mundo. Only got 1 skill shot, his build is semi flexible, his kit does well against tanks which are popular in the top lane right now.
Mid lane; Fizz. Everyone hates seeing fizz, he can snowball, roams well, and off meta builds are viable on him.
Jungle: Vi, good clear time, pretty simple kit compared tof other high skill cap junglers, very strong, flexible build path for either lots of damage, or more tank like.
ADC: Caitlyn, strong as always, can put some distance between you and front line, kit is semi-kite capable, can build to wave clear and fight tanks at the same time.
Support: Blitzcrank, Ranged supports are super popular right now and blitz can literally close the gap and his naturally semi-tank build comp makes him perfect to counter Ranged supports.
