How to become a professional design researcher?
I think that a Ph.D. program in human-computer interaction (HCI), such as this one, or a Ph.D. program in information systems with HCI specialization would be among the right routes toward your goal. Potentially, depending on what areas you're interested in, a degree, certification, postgraduate program or, even, a set of MOOCs in information architecture or information science might be a feasible option. An important point: unless you want to teach HCI at a university level, you can also IMHO consider trying to get your foot in the door without earning a Ph.D. degree - with enough knowledge and right skills, obtaining which should be doable, if you can self-study well.

In regard to performing research temporarily on your own, you can get a lot of ideas by searching for and analyzing information on past and current research projects, small and big, by individual researchers or small groups (review existing literature - peer-reviewed journals, conference proceedings, etc.; consider joining special interest groups (SIGs); visit relevant websites and blogs; consider professional networking by interest) as well as large and well-known research labs and divisions, such as MIT Media Lab and Human -Computer Interaction Institute at CMU. Good luck!