There are many different ways to play this game. In order to rank up, you have to figure out which play style works for you. Hopefully my post can provide ideas.
First - Pick a class
*While it's possible to reach solo queue legend with a tank or support class, I would not recommend that. It's going to take a long time to reach legend if you're unable to carry games. I've noticed from climbing, that the matchmaking will often stick you with four clueless team members against fairly competent enemies. These are winnable games if you play smart and carry your team. Tanks and supports can play smart, but it's tough for them to fully carry a game.
**When you do pick a class, stick to that class. Don't change. Even if your team needs a tank, don't trust any other person in solo queue with your role. I used to switch to tank if the team needed it, but then I realized the person playing my role was usually feeding, bad, or just not helpful. All the way through epic, I stuck to my class when we didn't have a tank.
Second - Gold
*I almost never lead my team in gold. Farming is not as easy for a caster as it is for marksmen, assassins, and fighters.
Instead of farming, be there for every team fight. Position yourself so that no matter what fights your solo queue allies get into, you're not far away. Unless you're a marksmen, staying away and farming while your team fights isn't really that productive.
Third - Playing in game
*You should be constantly watching the map. Know or theorize where enemies are the entire game. If multiple members of their team are missing, don't walk into brush alone. Don't go into enemy jungle alone. You have to be the patient one. Wait out their ambush.
**Positioning in team fights is very important. ALWAYS start towards the back. Let them enemies go in first. Once they commit, land your combos. If your spells are on cooldown, you should be repositioning. Never get too close to their team without having an escape off cooldown(like flicker). If you have an AoE spell, try to get as many members of the enemy team with that spell as possible.
***Let the enemy team make mistakes. Being patient is big in this game. If you wait for the enemy to get out of position, then you can make them pay for it.
Fourth - General Observations
*I noticed that it took the longest for me to get through Master and Grandmaster. Those are hard ranks to advance from, because you're often paired with teammates that feed, pout, or just don't understand the game. Try to stay positive during games. If you encourage your team members, they tend to play guilty. This is hard to do, and I would definitely get mad at my team from time to time. "WHY WOULD YOU DO THAT?!"
Fifth - Item strategy
*Unless you're playing somebody like Eurora who needs heavy damage burst, build items that can help your team. For example, when I played Cyclops I'd often get frost. This never helped my burst, but it did help slow their team so that my team could catch their guys out.
**If they have a good assassin, get immortality at some point. That ensures you won't be a target for their assassin and one shot early in the match.