It is ok, but it would be incorrect if you used what Karl suggests. I think that you don't mean "to study" Turkish as your profession. So, you can't say "okuyorum" instead of "çalışıyorum". "Çalışıyorum" is fine in your sentence.

you can also say; Türkçeyi kendi başıma öğreniyorum.

yes that's right. you could also use okuyorum instead of cailsiyorum.

Spelling mistake sorry... Türkçe*