First of all build confidence. Find one or two lanes that you just feel the MOST comfortable, regardless if you're comfortable anywhere, and strive to get that role. If I'm top 3 pick, I almost ALWAYS want to go top or mid, as those are my MOST confident roles, with jungle being me 3rd option. Of course if I'm 4th or 5th pick I fill. Also, since you're in Silver currently, it almost never matters what champ you use, as long as you excel with that champ. So find a few champs per comfortable role/lane that you just KNOW you're going to do good with. That's where it all begins. Confidence and comfort level. It's never good going into a game kind of on the edge about how you'll perform. You always play to win. While you're playing, feel out your lane opponent. Try to understand within the first 4-5 minutes of the game how he will play. Is he passive? Is he aggressive? If he's passive, he will be harder to kill for one of many reasons, a couple being:
A) He's relying on ganks to win his lane
B) He's not going to leave himself open
That means you'll need to ward, and take advantage of the farming, pushing your lane, and roaming to help other lanes if you're mid or top. If he's aggressive, then that means he doesn't care much about his jungle ganks, but that means you need to let your jungler know how your opponent is playing, so that maybe you can setup a gank for him. Wards are crucial any way it goes.
Once you've built an idea on how your opponent plays, you need to then have a strong understanding on what your role really will be. If you're top, are you going to be the primary tank, or are you going to be an off-tank/bruiser? If you're mid, are you an assassin, a fighter, or are you a mage? Once you understand that, you build accordingly. If you're losing lane, your first couple of low-tier items should be to counter the opponents offensive ability. Once you have all of that digested and you can do it all without thinking too much on it, then comes to harder parts.
To get out of Silver, you yourself will have to, more-so than naught, carry your teammates. To carry, it doesn't mean that you have have over 3/4 of your teams kills. To carry means you are: A) Setting up plays for teammates to safely secure kills, B) You are saving teammates from deaths by getting a kill, or assist, off of it, C) You're not feeding. You will have to start being more aware of your surroundings. The mini-map should be looked at every second you get as it tells you where the Jungler is, where your teams lane opponents are, where your wards are, where their wards are (if spotted as they drop em) and what camps are up. You will have to start utilizing pings. If you're mid lane, you'll have to start counter ganking lanes. The time to sit in your lane all day and have a tea party with your enemy while farming is over.
Lastly, you'll have to become a better decision maker. You have to decide, if leaving your half pushed lane, that is vulnerable to tower damage, open so that you can go bot lane and get that kill to advance your bottom laners and potentially take that turret as well, is a better idea than KNOWING your tower won't take damage because you're sitting in your lane, hoping your bot lane doesn't feed the gank that's about to happen, or the engage they are about to commit to. You'll have to decide when engaging is the right time and when not.
I remember I was stuck in silver on one of my other accounts. I gave up trying to get out of it, but I knew I deserved to be somewhere else, and knowing it's not going to just fall in my lap was where I started. This account was Bronze 3 a week before I hit Gold. Getting to silver was the easy part. Getting through Silver was even easier, but it was that Gold promo where it got difficult, only because I became a little intimidated by the simple thought of, "Hey, I might actually become Gold. My work will finally pay off." I started thinking about the future, instead of the present. I played nothing but Katarina mid, and Malphite top/jungle.
