I'm no expert, but I managed to climb from Bronze 4 to Silver 2. So I guess you should take what I say with a grain of salt but at the same time my experience should be pretty close to yours and insightful for your situation.
tl;dr: Your team-mates will be bad, it's normal, keep smilin'. Play solo-lanes, learn to ward, win your lane, avoid big team-fights, focus on objectives.
First off, the most extreme shift in my performance came when I realized this: if I really deserve to climb, it's normal that people around me are bad. There is no point in blaming them, you can't depend on them. If they were playing as good as you, it means you are at your right level and don't deserve to climb. So you HAVE TO carry yourself, and take the fact that you are paired with bad teammates not as a sign of bad luck but as a sign that you will climb sooner or later ;)
This won't help you win games directly, but it's very important to have the right mindset, stay positive and goal-oriented, instead of frustrated and tilted.
Second, I'd suggest playing Top or Mid. Basically, play a solo-lane where you don't depend on your team, because again, they are bad, it's normal, can't complain about it, just have to play around it. So no bot-lane where you risk being 1v3, no jungle where you can't depend on your laners reacting properly to your ganks.
Then, consistently win your lane. Like, more than 80% of the time. Until then, you can't worry about winning the game as a whole if you can't even win your own lane. "Winning lane" doesn't necessarily mean being 3-0 at 10 minutes, though. If you play a very hard scaling champ (Veigar, Nasus, ...), it just means being even in gold with your opponent when the outer towers fall. If you play Annie, on the other hand, just being even is not enough. It all depends on the champ, so understand your own champ and what "winning" means for them, but win your lane. One thing that is shared by everyone though: learn to CS well. You can't win lane if you don't CS properly.
Third, assume the enemy jungler will gank you, and that your own jungler won't help you. For two reasons: first, people are bad, you have to expect it (you start to see the pattern?); but more importantly, a jungler don't owe ganks to their laners, that's a common misconception. The job of the jungler is first and foremost to get gold, just like the laners. Ganks are often the easiest way to get gold, so that's why most junglers spend their time ganking, but a jungler's job is not to win your lane for you, nor is it to counter the enemy jungler. Also, if you are already winning alone (and you should), it's logic that your jungler will focus on something else, and that the enemy jungler will try to stop you.
So concretely, ward a lot, maybe tell your jungler if you see a good opportunity for them, but always play as if you were 1v1.5, and just take the ganks from your own jungler as a nice bonus, not as a due. Also, it's better to have no gank at all than a stupid gank where your jungler dives the enemy and end up giving THEM a kill, or even two if you tried to follow him. Do not hesitate to ping BACK your jungler (and of course, don't complain they didn't gank afterward ;) ).
Fourth: focus on objectives. Kills don't win games, objectives do. For real, I have won some games where my team has been 20 kills behind repeatedly. But ok, laning phase is over, you won, great. Now what do you actually DO to win the game? "Focus objectives" is a nice principle, but it's a bit abstract.
Well, at all time, you should ask yourself: "what objective can we take, and what can THEY take?". Really do it out loud, explicitly, it helps you focus.
- Before anything else, try to make side-lane push: that way, time is playing on your side, and even if something goes bad, chances are the enemy team won't capitalize on it and won't take too many towers.
- Once again: you cannot depend on your team-mates. So big 5v5 team-fights should be a last resort. So what else can you do except team-fights? Split-push or pick isolated targets, to create an advantage. Here, WARDING IS KEY, both to split-push safely and to find isolated targets roaming.
- Once you created an advantage, it's time to actually take those objectives, or force an unfair fight in your favor. When a fight is mostly over, ask yourself: "Can we take a tower now, or do we really need to chase the survivors?". If you can take the tower, go for it, kills mean nothing.
If you apply all of this, and if you are really better than the people around you, you should reach a 55-60% winrate relatively easily, which lets you climb a league in less than 40 games, approximately. Yet, it still means you lose 4 every 10 games. Losing some games is normal, you cannot win everything. Don't get frustrated.
Also, a very important point to keep in mind: this whole strategy assumes that you are better than the majority of people around you. Once you have reached your "true" level, this is NOT the optimal way to play.
A few more stray observations:
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I don't really like Trick2G persona, but is "My Way" series is very good to understand how to focus objectives. He also has this whole philosophy of "I'm better than my mates, it's my job to carry" that I emphasized a lot (even if he can be quite arrogant about it, which I dislike a lot).
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I suggested Top or Mid, but to be honest, I'm a Top Main and I never played well mid. Mid is more exposed to ganks even though the lane is shorter: warding mid is much more difficult than warding top, the champs played there are often squishier and the central location of the lane means more people will pay attention to it. I tried to play Support and Jungle. I think I'm a decent Support, but it's harder to impact games by yourself as a Support, since you are mostly an enabler for your team. And I never understood properly how to read a situation and react in early game as a Jungler, I think it's much more straightforward to play a lane and just win your lane. And yet, Trick2G is basically applying the same strategy I described from the Jungle, so it's definitely possible.
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I personally like splitpush a lot as a climbing strategy, but you can't do it with every champ, especially if you main Mid Mages. The core idea of split-pushing is this: you must be able to win any 1v1 against enemy team (or ignore them and take their tower), and some 1v2 (or at least stale the fight long enough). So the enemy team is forced to send "too much" resources to stop you, and the rest of your team should be able to win somewhere else, even if they are bad, because this is not a fair fight anymore.
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I'm less experienced with picking isolated target, but an important tip is: try to put yourself in your opponent shoes. Ask yourself, not what you want to do, but what they want to do, and then react accordingly to prevent them. A very basic example: if there is a big stack of minions on your side, what the enemy will want to do? Take the farm and/or depush. So don't stay with the wave, ambush your enemy on they way to the wave.
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Wave management is a bit technical and deserve a whole post of its own, but there are plenty of resources already available. Take an hour or two and learn how to do it, definitely worth it in the end, managing side lanes is important in late game to properly focus objectives.
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You cannot depend on your team-mates, but that doesn't mean you shouldn't communicate with them. If you intend to push and don't want your jungler to come, tell him. If you want to split-push, tell your team to leave you alone and go on the other side of the map. If a side-lane is pushing and you are too far away, ping it so someone else de-push it.
