Personally, as someone who made the climb from B4 to S3 this season (over the span of 2 months August to October) here's the advice I would give:
A. Tighten your champion pool: What I did when I started climbing is I went to champion.gg , I looked at all the junglers (jg main) and I picked 5. No matter what happened, I would only ever play 1 of those 5 champions. Rammus, Kayn, Poppy, Amumu, and Evelynn. I based these champions off of a few factors. Firstly, it's solo/duo queue, it's rare you get someone who wants to play tank, so I decided to pick up more tanks than carries. After that, I looked at champions who had solid winrates for players with less that 50 games played, and who's winrates increased with games played. This clearly gave me information that "these champions are easy to pick up, and as you learn the nuances to them as you play them, you will get better with them and win more." Finally I picked champions that fit my playstyle. For example, I am very good at macroplay, watching the map, team fighting, and dueling other champions. My KDAs are crap, but if I die, I almost always get a kill and an assist for it, and if not I'm supporting my team and picking up a lot of assists. This bleeds into my next point..
B. Find a Duo partner you can trust: This has always been a big one for me. If you're having trouble climbing on your own, it's because you're trying to carry 4 other players. If you have a teammate, you only have to carry 3 other player's maximum, but usually less because your teammate will be pulling his part as well (hopefully). Find someone who compliments your strengths. For me, I found 2-3 really good ADCs. Because I picked 3 tanks, most games, I path to level 3 gank bot, get my ADCs 2 kills, back and just repeat gank until they're super ahead and then I build Knight's Vow, and just defend/peel for them going into the mid game and try to win off them scaling early.
C. Change your queue times if possible: I've found out this season, that while trying to climb, all games after 11pm local (for me) on weeknights just don't work. There are so many college students who get stoned or drunk and then think they'll just hop on league and play. I changed my queue times from 10pm-1am to 6-10pm and it's worked wonders. That combined with the above tips really gave me good quality teammates in the right mindset for games.
D. /Mute all: Your teammates can communicate everything you need to know via pings. If you're in lobby, and people seem to be typing too much, just mute when you get into game. You waste time reading chat, plus if you're duo'ed you can be in voice with your teammate and you two can make calls together.
Hope this helps, from another Silver scrub. I'll be Gold soon though 