It's been a while since I've player in Silver (I was placed in Silver 3 last season and hit Gold 4 in 40 games), but here is how I got out.
The champions I played filled at least two of the following criteria:
1. Champions who could easily survive early game and had a very strong mid-game
These were chosen since mid-game was the most volatile time, which would decide whether a team won or lost. If you were capable of dominating this portion of the game, you could easily transition is to a victory.
Nasus and Irelia were especially good picks for me in this environment since they are quite capable of crushing most matchups if they don't fall behind early (and players in Silver often don't know hoetc.w to gank, rotate, or deny them farm properly) and split-pushing until the game ends, killing anyone who comes to stop them.
2. Champions with strong poke/waveclear and teamfighting.
I found that games would often devolve into both teams standing awkwardly beside objectives (i.e. baron, dragon, mid tower) for long period of time. Eventually, one team or another would get caught or sloppily engage a fight.
Varus (ADC) and Ziggs (mid) worked well in this environment since they could easily poke down the enemy team before fights, catch someone out, or disengage for their team. They dealt large amount of AoE damage, providing good CC, and had high waveclear/poke. They are were both also quite strong in the mid-game as well.
Gragas (jungle) worked quite well to since he could poke/waveclear (Q), engage/disengage/catch (E, R), had strong early ganks and good scaling, and could use his passive to mitigate poke damage.
3. Champions with CC.
You can see that all of the champions I listed above had at least one strong CC ability. Players in lower ELOs will often pick teams full of damage-oriented champions but many don't bother picking CC-oriented champions. You need some CC on your team to lock-down targets, engage fights, or disengage fights with any reliability.
I'm not saying that you should pick these champions in your games. What I'm trying to say is that you should try to identify common mistakes or trends in your ELO and figure out which champions you enjoy can take advantage of them.
Also, it may be a good idea to work on your item builds. While they aren't necessarily horrible, they could stand to be improved in a few areas.
i.e. You built Ravenous Hydra instead of Hextech Gunblade on Jax, Infinity Edge and Death's Dancer on Fiora instead of Black Cleaver against WW and Shen, etc.