It's really not that hard. It takes awhile, but the idea for solo climbing is simple: learn and play every champ.
Ideally, if you understand every champ better than whoever is using that champ when you face them, then you'll know how to capitalize, or if you can capitalize (hard counters do exist). It will also help you in champ select to better know who to ban, and who to choose in consideration with who your team has already chosen, or wants to choose.
You need to be able to outplay your opponent, and the best way to do that is to understand their champ better than they do. So try your best to play every champ, within your means.