How To Win Games, From Experience
Pick stuff that works.
Playing as the 4th or 5th damage-oriented champion on your team is a terrible idea and doesn't work. One good initiate from the other team on you or any of your hungry teammates will render you all useless. Instead, pick champions based around the assumption that your team will have enough damage. If you have two carries, it will. Don't aim for the most damage - aim for the most value.
Some champions like Lee Sin are value incarnate; so long as you know how to play with them you're never wrong for playing them. Cho'Gath and Malphite are never the cause of lost games. Any solo laner that can dish damage out as well as shrug it off is an amazing boon to any team. Supports can have incredible value as well - Lulu makes your damage-oriented teammates (which are everywhere in solo queue) incredibly overpowering so long as you are level 6. Taric past the 25 minute mark makes a game unable to be lost unless it already is. I do not know how to lose a game with many of the champions listed here without wandering away from my team and blatantly feeding.
**TL;DR: **If you ever want to win games or you just want some more breathing room, drop your assassin or hypercarry with 21+ offensive masteries and pick up some utility for your team. It works.