You don't really need to lean more champions until you get higher rank if you want to improve. You can know only a couple champions and improve on the basics of the game (laning, objective control, adjusting to your team/how the game is going, etc). I played from bronze to gold season 3, and twice this season (so rusty when I did placements for my accounts) and both times I played mostly one or two champs. When I started playing other champs I was able to play above bronze level even when I didn't play the champ much before just bc I had improved overall at the game.
Play what you love and try to get better at the game w/ those champs, you'll learn match ups and other champs through experience playing against them. There are a ton of one trick ponies in diamond, so if you truely master (holy shit that would be hard) one champ you can climb really high, it just may not be as much fun. Personally I burned out a bit on my main champ and started playing different stuff on the second account when I started climbing w/ that one.