Honestly, this game has really had balancing. While it is possible to win games with Evelynn it will take a lot more effort than winning games with Nunu. If all champions were balanced then it would not matter which champ you chose as long as it contributed to what the team needed and you were good with it. Right now there are certain champions that you can do really well with and they just won't win easily, especially if the enemy team chose a bunch of champs that are really strong right now.
For instance, I am a Nocturne main. He is extremely weak right now because of the current champs that are strong in the current meta. Because of this I started to practice Zac a bit, just for fun. In games where I play Nocturne I have to play damn near perfectly to carry a game. Games I play with Zac, a champ I just started learning a couple of days ago, I can mess up again and again and still end up winning because he is currently very strong.
Your best option is to get really good with a handful of champions and when your main is strong you spam that shit as much as you can. Once your main starts to get weak because of meta shifts then figure out what is strong in the current meta and have a few champions that can pick up the slack until your main becomes relevant again.