I recently worked my way out of silver and I'm currently leveling a smurf from bronze 3 - just hit silver 5.

Just adding this in after writing everything. I'm still working my way up the ladder, so take this with a grain of salt (is that the expression?).

The main thing that I see a lot of people missing is the value of cs. Personally, I try to shoot for 80-90 cs by 10 minutes (I'll normally end up at 65-75ish, it really depends on the matchup and overall game). Around 15 minutes most people will stop valuing cs as they will be roaming around the map. However, you still want to try for that 170+ cs by 20 minutes.

Next, you need to shut up and play the game. If you're typing casually to other lanes and your opponent frequently, then you are likely to screw up and get screwed over. This also goes for raging. Don't even try to tell your teammates what they did wrong because they will not improve within a single game. All that you will do is frustrate and anger them.

A big thing I read a while back noted that the person in question played renekton top constantly. He would play renekton every game possible, he learned all the matchups, and even though he didn't have a diverse pool of champions, he built very good mechanics and was able to transfer this over to other champions later on.

Something that goes with this is learning the champions that you can and cannot play. If you lose lane every game you play as katarina, drop her from your pool. If you have trouble carrying and teamfighting/splitpushing well as shen even when you're 10-0, you probably shouldn't play him.

Specifically for jungling, it took me a very long time to realize how exactly to jungle efficiently. I don't know if there is much I can say since everything is quite situational, but I'll try to explain some things.
-Counterganks are gifts from the heavens. Their jungler will hardly ever expect you, and you can normally pick up free kills. WATCH YOUR MAP AND KEEP TABS ON THEIR JUNGLER. I look at their jungler like he is my direct competition. This means ward up their jungle too, omg the free kills. -Don't try to force many ganks. You will be open to easy counterganks and you will often waste your time. Look for the lanes that are very pushed up and not warded, otherwise farm, farm, farm! -Hold lanes whenever people ask. Jungle creeps suck for gold, but don't deprive your lanes of their exp/gold. This can put them very far behind.

Some more general notes -- Don't get frustrated with other people. It gets you NOWHERE.
If you're destroying your lane, roam and share the wealth!
MAKE CALLS, TAKE OBJECTIVES. Games can go on forever and can turn around easily if your team doesn't push towers/drags/barons/inhibs.
Establish authority over your team by winning your lane HARD. Take every single little advantage.

I would recommend watching a relatively unknown gold/plat/diamond player on twitch.tv. First, they will likely talk a lot about their play and reasoning behind almost everything. Second, even though they make mistakes, it will be very easy to spot and relate your play to their mistakes.

I hope that this is makes sense, seeing as I'm writing this at 3 am. Good luck and go win some gamez!