What is the best way to filter Turkish coffee?
I think you got something mixed up there. It is normal to have noticeable particles while drinking Turkish coffee. Everybody uses a strainer for it, and it does result in a slightly "muddy" drink. 

I guess you could pass the prepared Turkish coffee through a paper coffee filter before drinking, but this would be a very unusual way to do it. People who don't want to taste the particles just use a different brewing method in the first place. Making it with particles and then removing them is twice the work, and I can't imagine it giving you any taste advantage. 

If you do any filtering, the coffee will end up less strong, both in taste and caffeine. It is the particles you taste which are the actual coffee. Another method which achieves strong coffee is true espresso. It needs an expensive machine which passes the water vapor through the grounds under pressure though.