I don't believe there is a way to be totally private - the best you can do is plausible deniability.
1. transfer USD to exchange
2. buy BTC
3. create (throw away) account on blockchain (with disposable email address) / or start with blank profile in bitcoin wallet application
4. transfer BTC to previously created account
5. repeat steps 3&4 many times, splitting amounts between accounts
6. create final account(s) and transfer founds onto it
7. erase all information needed for executing steps 3-5
This way there is no direct link between your money getting into exchange and final account.
The intermediate addresses could be you, but it may as well be X other people/places/organizations - plausible deniability ...
Just hope you will do something good with this - like
supporting Wikileaks - and not something illegal ... right?