I can only tell you what I did, and what I'm continuing to do. I spent all of Season 3 in Bronze (Bronze V quite a few times), and made it into Silver just as Season 3 closed. I'm currently in Silver 2 and climbing. I'm no pro, but I was very careful about how I did it, so I'll explain.
First, those chumps who tell you to "carry by yourself" are terrible teammates and often high-level players who have smurf accounts. They may be able to carry a game by themselves, but I've seen them fail many times and rage about how bad the team is, so even they can't deny the importance of teamwork. They just think they're good enough to carry the weight of 4. Rarely is that true. Anyway, enough of that arc (those guys really annoy me, if you haven't realized).
1. Communication - In every lobby I'm in, I try to get people talking about what they're going to play. I never tell them who they should play, because I'll take a loss over being a jerk. I will suggest what our team needs to be more well-rounded. Some take that as an order and get upset, but this is very rare. What I've found, ultimately, is that people are very accepting of someone opening up communication, and teams often get very serious about building a solid team comp. Team Comp is key, but the communication also slides very well into in-game teamwork.
2. Your Role - I used to play mostly mid and ADC because I thought they were the most fun. High Kill Rate does not equal a victory. Instead, I started focusing on filling the void our team needed. I learned the strong champions, learned to love Support. I improved my Jungle role. At this point, there is no lane that I can't play well enough to play it in Ranked. Currently, I try to play champions that control the flow of team fights (tanks). This way you can choose when to engage and hopefully the team will follow your lead. I've found that softly reminding people who they should focus before the game starts works out so well. Something like:
As long as Akali makes sure that Jinx dies, we've got this easy.
3. Study Yourself - Use LoLKing.com to look at your stats. Find out what champions you are good at playing consistently (Warwick appears to be your strongest so far, but you don't have enough games to know for sure yet). See what you need to work on, and improve that specifically in AI and Normal games (always go Draft Normal so it's closer to a Ranked setup).
4. Galio - Maybe he won't work for you, but that's how I climbed out of Bronze. Galio completely wrecks most AP Mid champs, he's tanky and his ult can dominate a team fight if you use it at the right time. You can build him as an armored Tank if necessary, or as a resist Tank and he's extremely useful in both situations. His damage output in the right conditions are top-of-the-charts and strong even in the worst games. Maybe it doesn't have to be Galio, but a champ like him is what you need. One that can turn the tide of a team fight in an instant.
So, I know it's a long response, but this is how I got out of Bronze, and the same method has me quickly gaining on Gold at the moment. I haven't changed a single part of my strategy. I've cycled my champions around, but everything else is identical, and it's still working. I played Gold-ranked players regularly now, and I have no trouble beating them when the team is working together. I still have games where people don't want to communicate, but those are rare and I'll still pick what I think is best for the team, no matter what.
The best lessons I learn, though, are the games I lose. The champions that dominated us, the way the other team was built. I study everything and adopt the things that worked. I got tired of getting beat by Shyvana, so I learned to play her and now I'm beating people all the time with her. Learn from losing.
