I'd like some help with my CV. I want to add one sentence below my name, telling the company that I am free to relocate to any city. I am not a native speaker and I am not sure about this. Can anybody tell me how to say it? Which way is proper and professional?
Your question actually has a sentence in its body which would work: "free to relocate". It's not a complete sentence but CVs often use fragments such as that for brevity. You might also say "willing to relocate", etc. @Robusto's "will relocate" works too but it sounds to me like you are saying that you already know that you are relocating at some point in the future.
Address the issue of relocation in a cover letter, rather than your CV, and tailor the description to the particular requirement of that position. A résumé or CV is a description of your previous experience, while a cover letter is where you should promote your suitability for a particular job - including requirements like relocation.
If for some reason it is absolutely impossible to provide a separate cover letter, consider adding a general Availability section with additional information such as dates you can begin working. "Willing to relocate" is a suitable phrase.