The trick is very simple, it is to not die. I know that might sound very sarcastic, but its the truth.
I am currently in the bronze division, and I watched some of my replays to help improve, and I noticed how many times we simply fight for absolutely NO REASON. Even after we fight and one of the teams have been aced, we simply go back to base and get more items. For what? We already killed them? Why do we have to prove we can do it (or worse NOT do it) again?
I have watched some higher elo games and in some games they dont get first blood until 20 minutes in. 20 MINUTES IN A GAME FOR ONE KILL. After 20 minutes in a bronze game its 49 to 54 and the jungler hasn't even looked at a single drag.
So I started paying attention and choosing battles when they are in my favor, and believe it or not I win more fights than i lose. Even when I am in a lane where I have been counter picked, as long as I don't feed the enemy and let them snowball, I AM WINNING MY LANE (of course cs is important too).
So when you look top and notice the 0/9 darius on your team, you simply can't stop the 9/0 renekton coming for you. What makes this Darius think "man, next time i will get him for sure"? It is very hard to solo carry when others cant understand that not dying, is just as important as getting kills.