I've been thinking about this for quite a long time. When I reach an advanced level in your language, I find it harder to improve. What can we do when we'd like to reach a higher level (beyond advanced)? I wrote 'from C1 to C2' in the title but honestly, these are only parameters and they are not really representative. I mean, even if you pass your C2 exam, it doesn't mean that you can speak as fluently as a native speaker. What I want to illustrate is the point of transition from advanced to really fluent/native level. In my case, it's English. I find my writing and speaking particularly weak. What I am doing now is trying to write more, doing shadowing exercise, and making mini impromptu oral presentations.
Do you have any thoughts/even methods to share :)?
First of all, you sound really good already! Your English skills are excellent.
If you want to be really fluent, you'll probably want to learn slang, idioms, and colloquial speech, and make sure your accent sounds native.
For speaking, be sure to talk to lots of natives. Have face-to-face conversations or Skype them. If they can tell (by your accent or something) that you are not a native speaker, they might be tempted to talk to you slower/more clearly and avoid idioms. Ask them to speak to you like they would their own friends, completely colloquially. Visit English-speaking countries if possible and immerse yourself in the language and culture. Put effort into pronouncing things with the correct accent. Watching movies and listening to popular music is a good way to increase your fluency as well.
For writing, try to write at an essay, article, or research paper level. If you're really interested in writing, you could try to break into the freelance market and write for websites or blogs (you'd have an interesting perspective for language-learning sites).
Hope those suggestions help! (And btw, like I said, you're writing skills are very good now) :)
Personally I found short stories very helpful, both reading and writing them. There are websites where anyone can share their writing; reading that you'll be exposed to all kinds of "natural" English; and although the sites usually recommend that writers edit their own work before posting it, you'll find many young but also older native speakers with bad style who make typical native speaker mistakes in English; it's good for your ego when you can recognize those mistakes ;) And you'll learn a lot of pop culture.
Finally, if you have a passion for writing yourself, you can post your own poems or short stories. With some luck, you'll find a kindred spirit willing to edit your work and give you feedback on your writing.