I'm going to say, based on what I've been doing last couple days(went up 200 elo in 1 day(23W-3L)), is that you cant just have the mindset of "I'm the best, I'm going to carry, KILLS, KILLS, KILLS" because thats how a lot of people are going in with that mindset, and tbh look where they still are. The thing that I've seen and have tried that works the best is to play a tankier champ that has cc and can still output good damage(things like jax, j4, darius, panth, ww, sion, olaf, riven, poppy, so on so forth). This is because these champs have the fixes for certain things other roles lack.
AD carry, very squishy and easy to be focused, for the most part low to no cc, can be destroyed due to lane partner being dumb.
AP carry, for the most part very squishy, almost all have 1 burst then they dont have any damage possible to kill everyone else, easily focused, fall off late game.
Support, if your trying to carry this wont do you any good unless you have a good AD carry(judging the fact your trying to get out of elo hell this is very unlikely), almost no dmg, cant carry.
Due to this thought process, your best bet is a jungler or a solo top.
Reasoning: Not very easily focused down, moderate/high damage output, hard cc (DONT TRY TO CARRY SHYV), damage that can last an entire fight (not just burst then done), Saving capabilities (can jump in and peel or save a team mate), quick objective kills, can easily get team fight items(aegis, zeke's, etc) without sacrificing too much damage or tankiness.
Mind Set You Should Try To Use:
-Staying alive is more important than getting a kill
-saving a team mates life is more important than chasing for a kill
-Push objectives (dragon, baron, red, blue)
-If your down by a bit DONT fight unless under your turret
-DONT BUILD GLASS CANNON
-Focus the carries(DO ANYTHING to get to the carries(flash, ghost, cc, spells))
-If you ace the enemy team, apposed to running straight to baron, you should attempt to down a couple turrets and then run to baron when they're under 10 seconds until they spawn.
-DONT BE THAT PERSON (we all know one, I'm talking about the people that get mad at other people and decide they're not going to help that person at all(yet in all reality that's you just making your team lose))
-Let your teammates know when your going to do something or if they should follow
-If you notice somebody doing something wrong, just nicely explain what they should be doing
That's my opinion on the best way out of "Elo Hell". If you think I'm wrong, PLEASE, tell me a better way and why it's better. In addition, check out my steam "The Very Depths of Elo Hell" and me attempting to get out at http://www.twitch.tv/sycosoldier90 (I've gone up 300 elo in the last 3 days)