First thing's first, one of the best ways to climb soloqueue is to main one champion and be really REALLY good with that one champion. If not two; one of the best ways to carry is to stick to one really solid pick for Mid or ADC. I'm currently working on my mid and adc game because I'm tired of following the rules and doing everything right as a support/tank jungler, and losing because some idiot picks Ashe, goes 0-20 to the enemy Vayne, and we lose because Vayne can now 1v5 the entire team. If you're not a good adc, watch videos, play regular games until you carry almost constantly. If you have trouble getting there, you know how to adc but get outplayed, try practicing in bot games. I've been playing a bunch of games as Jinx just to get good with her. I noticed that I needed to work on my reflexes, so I practice against the intermediate bots. The beginner bots are awful at pretty much everything.
Sometimes, people think they know how to adc like they're diamond, but they're missing certain tricks to the trade. How is your last hitting? How well do you poke? Do you know your match ups well? Do you know your general damage outputs and power spikes? Your opponents? Can you kite/animation cancel? How is your positioning in teamfights? How do you adapt your build to the situation? Do you control objectives correctly? How often do you buy wards and place wards?
These are just a lot of things that I noticed that I didn't do, and have to work on still while I try to "break out" from bronze elo to where my favorite champion to jungle with, Sion is a more viable pick (since, in bronze, your success as Sion is pretty much directly connected to how good a player your adc/mid is).
I also suggest (for soloqueue) you pick a very aggressive champion. If you prefer to top lane, pick someone like Nasus or Renekton. If you're ADC, pick someone like Graves or Vayne. In mid you have all kinds of champs that are really aggressive like Xerath, Leblac, Yasuo, Nidalee... Just don't pick someone who gets banned often like Katarina. In the jungle you have a few picks like Nidalee or Hecarim that are pretty strong, and very aggressive. Yi is a bit out of favor, but you're good with him he can be pretty scary.
You can carry as a support main, but your problem (From what I can tell) is that your teammates aren't that good, so I don't advise it, but if you do try to pick someone like Blitz or Thresh. Morgana requires a bit more skill to land her apocalyptic "you're not making it out alive" skill, so you may not want to pick her at first.
I hate to say it, but in ranked at low elo, you're not always going to get a good carry, but you can eliminate this problem by being the good carry. I hate to leave it on a note like practice to get better, but that's essentially what you need to do at your choice of adc or apc. Eventually, you're probably going to need to be skilled at all roles anyway, so now would probably be a good time for you to pick up a carry role.
