Howdy, I started playing in s4, and ended up in silver 4 oddly enough. s5 my peak was d5, and ended the season in plat 1, and s6 I'm currently sat comfortably in diamond 4. I just wanted to share a few tips, some general, some aimed towards the support role which I main to help those looking to climb. Can't promise anything, but these are some general tips I've found along the way.
1) Focus on your own game. Don't blame others for your mistakes, don't care what others are doing. You focus on your lane, you try your best, you win your lane/role and do everything in your power to win.
2) Don't join in with petit arguments. Ignore the people who are flaming/moaning. Can mute them if you'd like, can communicate fine with pings. Numerous times I've caught myself typing something which would have just sparked things further, before realising it would be stupid and deleting the message. Don't provide fuel for the fire.
3) Team comps matter more than people think. I'm someone who will generally try and pick a champion that the team needs. If we have a squishy top/jungle, I might pick Alistar to give the team some front line and tankyness. If the enemy team has Rengar, Riven, Talon, I might pick a Janna/Lulu/Karma/Braum to try and peel for my back line as they have a lot of dive. Preferably don't go into a game thinking 'I'm going to play X', as X may be a godawful pick that may lose your team a game.
4) Don't pick up the FOTM champions without practice. You're much better off with your tried and tested pick rather than picking up a new pick in a ranked game because you saw it played in LCS. Soloq is not LCS. Saying that however, there is most likely a reason why champions are picked up (waveclear, overtuned numbers, overloaded kits, utility) so feel free to practice in normals first. You can generally get the hang of a champion well enough within a few games. You obviously will not master the champions (knowing exact mana costs, exact damage capable of doing, ranges of abilities etc), but can likely play them well enough.
5) Try and win every single game. So, it sucks, you won't win every game. But I've won games being 9/10/11/12k down, 3 inhibitors down, and somehow manage to make a recovery. It's that one 'no' vote on the surrender, or that 1 person trying to keep the team's attitude up which gives you that 1% chance to win that game. And that small chance there can make a difference between a won promotion and a loss.
6) Know when to stop. I've experienced it a lot this year - win streaks and loss streaks. If you've lost 2 games in a row, chances are it isn't your day and it's good to stop. If you've won 3 games in a row, it might also be a good time to stop as good games don't last forever. Take the +70lp on a day, and don't risk it on that 'just 1 or 2' more games, which could put you back on your original LP.
7) Almost every single death/mistake in League is to do with being greedy. You need to be looking at the situation you're in, and ask yourself 'is this stupid?' if you're sat in lane with 200hp, 18 mana, no summoners and no wards - yes, that's stupid. Back away, play safe.
8) If you get an 'off role', you're better off playing something with low mechanical skill and high reward. These champions still exist. If you've played jungle 8 times in your life, and get jungle, don't be picking Lee Sin, Shaco, Elise etc. Pick the easier thing, grab the Sejuani, Vi, Gragas etc. No shame in picking a 'low skill' champion if it helps you win.
Support specific hints:
1) If your AD is a cabbage, leave them. Let them sit in that lane and derp the hell out while you go and try and help mid/top or even the jungler with ganks, invading, vision. Numerous games won recently after realising that the AD was a potato, gone and spent 5-10 minutes helping lanes get kills, providing vision, helping get towers etc.
2) Expect the entire enemy team to collapse on you. Respect the enemy jungler's gank routes, respect the teleport, and keep an eye on that minimap for any MIA's from top/mid.
3) Know the bot lane matchup. Learn which ADCs are strong at what times. If you have a Vayne into a Lucian, know that the Lucian has a stronger early lane phase, so the Vayne can't contribute as much. Know that the enemy Bard support has the huge advantage over an Alistar at lvl 1 and you need to play passive.
4) On 'every' support (bar stupid things like velkoz, brand etc), the carries are your responsibility. If you're on Morgana support flashing in to the enemy to ult, and leaving your AD on their own, you're doing it wrong. You helped them through lane to get to this stage, now you keep them alive while they dish out the damage.
Hope it helps! Feel free to ask any questions here or ingame, EUW name MorroWtje
As an edit: I've done some coaching previously for anything up to challenger teams, so if anyone feels like they'd benifit from some, feel free to hit me up.