I've had to climb out of bronze every season I've played in (4, 5 and 6); provisionals in new seasons always dump me in bronze.
Randomonium gave a lot of good advice in his answer. Let me stress this additional point, and I'll note it comes from my challenger coach:
"Don't die. I know it sounds stupid, but really, it's the most important thing. If you're dead [I'm an ADC], you're not dealing damage or getting gold and XP. If you're not getting gold or XP, you're falling behind. If you're not dealing damage for your team, you're not fulfilling your role. Worse, you're giving gold and XP to the enemy team. So it's really, really important not to die."
Another important note: if your opposing botlane gets a kill or two and they're ahead, the answer is not to try to kill them to catch up. They're ahead, and will have an easier time killing you. Farm, and wait for them to get overconfident and mess up. THEN kill them. If they never get overconfident and mess up, they might be the strongest people on the opposing team. In that case, stalling them lets the better people on your team get ahead of the worse people on their team.
