I'm ending this season at Silver 3. I ended last two season in Silver after climbing out from Bronze 5. Even this season I started in Bronze 5. So if you want to know how to climb out of Bronze I can definitely help.
Firstly. All that high elo stuff you hear or read about, things like rotations, pressure, lane priority and all that, throw all that away. That is what you learn when you want to play League of Legends. No one is playing League of Legends in Bronze and Silver. They are playing some kind of shit where they compete to see who can feed more and who can be the biggest idiot.
That being said, the one and only thing you want to be thinking about each and every game in Bronze and in Silver is, "how do I kill everyone myself so me and my team can get objectives". That is all you should be learning how to do.
For an example I'll use myself. Last two seasons and this season, I mained Kayle out of Bronze. All I did every game was ignore my team early and mid game and try to farm up her powerspike which are these items:
. After getting those 3 items, I start grouping with my team with one single goal, to be the reason that they win all skirmishes and team fights. Without me, those idiots are going to die and give up everything. I'd try to position myself to get off as much damage as possible and on as many people as possible, often getting double and triple kills. After killing off the enemy team, ping whatever objective makes sense, whether it's a tower, Baron, Dragon, whatever. Get it, reset. Rinse and repeat until you and your team are at the enemy Nexus. It's that easy.
So find a champion or champions you like where you can apply this strategy. It works best on high damage hyperscaling champions with a knack for killing off multiple foes in one fight. Any marksman and some mages like Azir or Cassiopeia can work. Duelists like Trynamere, Fiora, Yi, Jax, Kayle and Yasuo work best. Bruisers like Nasus and Darius can somewhat work but you have to end the game faster with them since bruisers are generally outscaled by many champions. I do not recommend champions like Zed or Fizz since they are single target champions, who excel at killing a single target quickly. You would need your team not to be idiots that still die after you go in and kill their biggest threat. Zed is pretty much spent after R+E+Q, but with Yi you can keep right clicking until everyone is dead, Jax too and he has very low cooldowns so he can use his abilities multiple times in a single skirmish. Not much Zed or Fizz can do after blowing their load on a single champion so you better hope your team can clean up after you.
One final note. This is not the only you can climb out. You can climb out on any champion in any role, even Rek'Sai but this way is the way that worked for me so if you're like a support player or a tank player maybe someone else can help you but with high damage scaling champions that can apply their damage to multiple champions fairly easily, this is pretty much how you get out of Bronze with them from my own experience.