Getting to Plat. [Big wall of text guide]
I've been playing on a Silver 5s team - resulting in me playing against a lot of low Bronze/low Silver opponents as well as spectating my Silver teammates play. I've taken notes on the mistakes they frequently make and have come up with some advice on things that almost all Bronze/Silvers do.
Follow this advice and you'll get Gold. Follow it consistently and take it to heart and you'll get Plat.
Most low-elo players can actually last hit properly. They miss so much CS because they are focusing more on attacking the enemy player than on last hitting. Avoid forcing trades. Only attempt trades when the enemy is attempting to last hit and you aren't going to miss a CS by trading with them. Keep in mind AoE spells that go through minions that can allow you to trade and last hit at the same time. This happens less often as you climb rankings, but is very common in Bronze/Silver.
Forcing your opponent out of lane is almost as good as a kill. Don't blow a lot of summoners or ultimates to pick up a kill. If they are already being forced out of lane - you can get a CS and Level advantage on them. Then you can use your ultimate when they return to lane to force them out of lane or kill them. The result is you get them out of lane twice - instead of once (and then waiting on your ult/summoners to come back up). 2 > 1.
Ideally - your buffs should have minimal downtime. To prevent them from being stolen and to maximize the % of game a player on your team has a buff. If your Red Buff has been untouched and idle for 4 minutes - you're doing something wrong. Once you take 2 of the first tier turrets - try to keep the enemy buff warded either top or bot (assuming mid is one of the two turrets taken). Red is easier to ward by warding the bush across from it.
Learn how to kite (and when to kite). Silver and lower players simply run away as ADC's instead of kiting backwards. There are many times that they have died or could have turned a fight around if they were properly kiting instead of running away without attacking. Properly cancelling your auto attacks to chase is also important and not something I see often at these lower levels of play.
Use champion.gg - follow the runes/masteries closely for "most win %" so long as "most win %" is >1,000 games. There is a reason those runes/masteries have the highest win %. The next time I see 20:10:0 I'm going to cry. 3% damage > 2 MR guys. One of the biggest criticisms of Bronze players is they have terrible Rune/Mastery setups in general - let alone for the champion they are playing. (We won't get into their builds that "totally work"...)
If your jungler doesn't have smite up - if the enemy team isn't 100% visible on map do not do Dragon. If your Jungler does have smite - make sure the enemy Jungler is dead or Top lane/Jungle. This will make sure you don't get Outsmited and lose the objective.
If your team is far ahead - keep vision and try to group. Travel as a group to make picks in other lanes and force your way down turrets. The exception to this is if the enemy team has really good waveclear. Send your best duelist to splitpush and don't force a tower as a group until 1 or 2 people leave to go answer the splitpush. Too many times Bronze/Silvers will force a fight under tower before anyone leaves to answer the splitpusher. Causing your team to lose the fight and throw the lead.
If you are behind - start building your defensive item and play defensively. There are very few exceptions to this rule.
Try to take Scuttle crab whenever you have control of the river/enemy jungle. It's free gold and free vision.
Always ping the enemy jungler if you get vision of them. No matter what lane you are in and no matter which lane they are ganking.
Pick 2 champs in each role to perfect. One that is strong/often banned and one that is still pretty strong, but not banned as often. The champions should belong to different team compositions, so if both are available you can pick the one that best suits your team comp. Some examples for each role for the current meta:
Sejuani (Tank AP) + Rek'sai Jungle (Tank AD/Bruiser) Jinx (Chase/Engage) + Kalista (Kite/Disengage) ADC Maokai (Tanky Peel/Re-engage) + Hecarim (Tanky/Dive) Janna (Disengage/Peel) + Thresh Bot (Let's be honest, everything) Orianna (Utility Mage) + Cassio/Xerath/Viktor/Leblanc (bursty mages)
For your favorite 2 preferred roles - pick a 3rd champion. If you're really bad at a given role and would rather not play it at all, don't pick any champs. Just dodge if you get stuck with that role. It's better to lose 3 LP for dodging and try to get a role you are good at than suffer 7-22 LP for a loss playing a role you're bad at.
If your team has 0 tanks and the enemy team has 2-3 tanks. You should probably dodge, you were out-comped. Not to say it's impossible to win - but it's extremely hard after the 30 minute mark if your team isn't stomping their faces in.
Be conservative with your summoner spells. Avoid flashing if you are going to die (eg: caught out 1 v 4). If you burn a spell and are already behind, it allows the enemy team to be more aggressive and punish you. A wasted flash is 5 minutes of "kill them because they don't have flash". Likewise - punish enemies who waste their summoner spells/ults.
Know the enemy damage output and when they return to lane - tab and see what they purchased. Did the Annie leave and come back with a Needlessly Large Rod? Probably looking to all in you herself. Did she back and come back with T1 boots and a Fiendish Codex? Might be fishing for a Jungle gank, using the extra speed from boots to land her stun. Knowing what items they returned with lets you gauge their damage output and intentions.
If the enemy went back to buy - if your lane is shoving towards them already, should shove the lane as fast as possible and back. This denies them gold upon their return and resets the creep wave or causes it to push in your direction. If the wave is hard pushing in your direction, kill a few creeps to balance the wave out and cause it to slow push in your direction. If done properly, when you return to lane you can freeze the creep wave outside your tower.
Speaking of balancing/freezing creep waves - learn how to do this. It gives your jungler an opportunity to gank, can deny the enemy CS, and makes you harder to gank for the enemy jungler. This video can help explain things (was just one of the first videos in search results, I'm not affiliated with the website in any way, shape, or form): https://www.youtube.com/watch?v=2TUfTCfX02g
If you don't back when your opponent does - and they weren't forced to back early - they'll come back to lane stronger than you. This allows them to bully you in the creep waves and under tower, without you being able to fight back. This is what causes you to stay in lane at 1/2 HP being bullied under tower begging your jungler to gank before you inevitably get dove by your lane opponent and/or the enemy jungler. 8 out of 10 times this is the reason you're getting dove under tower.
When you back at level 9-10, if you have 250g, upgrade your trinket. It's one of the best early investments you can make. The 2-ward yellow upgrade is essentially a 250gold Sightstone. ALSO MAKE SURE TO USE YOUR TRINKET WHENEVER IT IS OFF COOLDOWN. Even if you never upgrade your trinket, in a typical 20 minute game you should have 10+ wards placed unless you're a Jungler and have a Red trinket.
Ward liberally. Expire wards you don't need. Try to always keep 1 ward in reserve (in case you're chasing through brush or need vision of a nearby bush before a teamfight). The other 3 in your sightstone should always be placed down somewhere. If you don't have 3 wards out, you're missing out on vision. Vision that could save your teammates or net your team a free kill.
There is a lot more to learn. Positioning, "guessing" the enemy jungle location, learning how to team fight, when to tower dive, when to group vs when to splitpush, who to focus in fights, I could go on for a while about more broad game mechanics. If you find yourself dying >8 times a game often - you need better positioning. I can't really help you with that other than "play further from your enemies, stay behind your tanks".
Tanks and initiators are exceptions to the 8 death rule, so long as your team is winning the teamfights.
Note: the 8th death rule is actually a 5th death rule. But in Bronze/Silver your teamfights will generally be less coordinated and result in higher deaths. I frequently have 7-10 deaths as opposed to 3-5 deaths in Bronze/Silver games that aren't a complete stomp in my teams favor.
I wrote this first and foremost for my 5s team. However, I feel many players could benefit from the information. I hope it helps you.
Vayne 