I've written once before about climbing the Elo ladder. That was nearly 10 months ago, and I wince on rereading that guide. I got some basics right--keep calm, communicate, comp well, and so on--but I had the perspective of a new-to-Ranked, barely-1300 player, and it shows. It shows most in my emphasis on picking a champion that can carry you out of Elo Hell. Since then, I've had it ground into my bones that the champion doesn't carry; the player does.
As before, this guide is directed at people who feel their skill level is above their Elo. I'm at the low end of gold, so I can't exactly advise 1800 players on getting better. But an 1800 player who thinks he should be 2100 is really in the same boat as the 1300 player who thinks he should be 1600, or the 900 player who thinks he should be 1200. In each case, the player perceives that he is not held back by his skill, but rather that his Elo perversely holds him back from rising, in the form of uncarriable teammates. Here I will lay down some principles that help you turn your team from a hindrance into an asset that helps you carry your way up the ladder.
(Some games can't be carried. Accept it and move on. You don't have to win every game to go up; you just have to win more than you lose. Finding ways to blame your feeder teammates for the last game you lost isn't going to help you do that.)
You aren't communicating enough. Yes, you. If that chat box isn't being used within the first 3 seconds of champ select, and isn't full within the first 15 seconds, you're not doing your job, which is coordinating with your teammates to find a comp that works for everyone. One thing that tremendously frustrated me in the 12-1300 brackets was when people would keep quiet until the end and then say "Oh by the way I can't jungle." The earlier you can get that person to own up, the easier life will be. Inversely, if someone has a main, getting them to say so can prevent role redundancy.
Speaking of mains: Have a few. Don't have just one--then you can get banned/picked out, or someone might fill your role before you can talk (see above about communicating), or you might get counterpicked, or whatever. But don't try to have two dozen--then you won't have sufficient mastery of each champion. The ability to play all champions to your Elo is not going to raise your Elo; being consistently better than your opponents with the champion/s you choose is what will do that. You should be comfortable with as many champions as possible, especially in key-but-unpopular roles like jungler/support, but focus your efforts on a smaller group. If your main is someone common, you need at least one more in the same role. Don't count on getting Singed every game.
Understand matchups. Part of mastering a champion is knowing when not to use it--so even if you're a pro Teemo, don't pick him against Mordekaiser solo top. Recognize what your goals are in your particular matchup. Don't focus on trading harass with Morgana at level 5--keep the lane off your tower and kill her with burst later. Don't fight Mordekaiser in your creeps--burst him from a distance. Don't let your team enter game against Tryndamere without Exhaust, against Twisted Fate without Teleport, against Shaco without CV.
Realize who has to die and who has to live. If you're jungling and the enemy has a Kog'maw, you gank his ass and shut him down. If your team's Tristana is 6/1 with IE-PD at 20 minutes, you protect her in teamfights even at the cost of your ass, because she will win the fight as long as she's alive to DPS.
Realize the point of winning fights. No, it's not your KDA. It's the ability to take objectives unopposed after fights. With that in mind, don't chase 3v1 just because he's not dead yet if you can take two towers instead. And don't go looking for a fight when there's no objective at stake. You should never get caught at dragon cave when dragon's not up, and neither should your teammates, so spam that Ctrl-click when necessary. The corollary to this is that if you can take the objective without a fight, that's better than fighting, so don't towerdive when you can slow-push and poke a tower down.
Maphacking wins games. Buy a ward every time you go back. If you're the support, buy three. Learn the good ward spots--next to enemy wraiths, between the river bushes, in front of dragon, behind baron, and so on. Realize what the threats/opportunities are at different points in the game and ward accordingly--don't just drop it in the nearest brush.
And back to the original point: it's all very well for you to do these things, but you'll win far more if you can get your team doing them too. That means telling them what to do in a way that makes them feel good about doing it. "Tristana, you're our carry this game, we need to keep you alive, and you need to stay alive." "Their only chance of coming back is getting picks or BDing, so push lanes and force Baron." "We can come back as long as we don't get caught in bad fights, ward baron and turtle." Pings are your friend. Ping the next objective. Ping where you think the enemy is. Ping to back off, ping to go in. Having a second type of ping is the best thing Riot has done for solo queue in a long long time. Make use of it. Carries not following you in? You're not pinging enough. Support getting caught out of position? You're not pinging enough. The more you can flood the chat box with GOOD communication, the harder it is for Negative Nidalee to start the blame game. And the more you give your teammates good direction, the less time they have to do something stupid.
The corollary to the above, of course, is that you absolutely cannot start the blame game yourself. Especially late game, anything bad that happens is the whole team's fault, because the team has to function as a unit to win. Raging is the surest way to break down team coordination, which is the surest way to lose. Don't do it if you value your Elo (or your account, for that matter).
Finally, realize that your skill level isn't high enough. If you aren't on the front page of the ladder, you have no reason to sit back and say "I'm good enough, it's up to my teammates to get carried." And if you're good enough to be on the front page, there are other concerns. I will hazard to say that nobody has plumbed the depths that League of Legends has to offer. Rock Solid gave us a glimpse when they shocked the world this past weekend by knocking off CLG and TSM in quick succession at the IEM NYC qualifiers with comps that defied the established metagame. We can't all hope to imitate RS' success--nor should we attempt to imitate a team that prides itself on being inimitable and innovative--but we can surely improve ourselves, elevate our games, and play better than we ever did before. At the end of the day, that's what rising in Elo is all about.