How do I become a flight test engineer?
Go to college and get an engineering degree, preferably in Aerospace Engineering or Mechanical Engineering with an Aerospace emphasis.  You may also want to pursue a Masters in this area after your Bachelors.
Seek out the best schools you can get into; the biggies in aero are Georgia Tech, University of Michigan, Purdue, University of Washington, and MIT.  But there are a lot of other schools out there with fine aero programs as well and attending there doesn't limit your job opportunities.
Study hard and get good grades.  If you can't hack your freshman and sophomore years, engineering may be a bad choice as a field for you; seek a different major.  On top of that, I strongly suggest you get involved with extra curricular engineering projects or clubs in your department.  This will give you a lot of hands on engineering experience and help build up a knowledge base for you of how to be a good engineer.
I'd also get involved with either a RC airplane club, or better yet, join a chapter of the Experimental Aircraft Association and participate in the flight testing of home-built aircraft.  Again, it starts giving you a lot of hands-on experience in the flight testing of airplanes, which looks really good on a resume.  You can also get a couple of good references this way.
In your junior or senior undergraduate years, start looking for co-op or internship opportunities at the major OEMs, i.e., Boeing, Northrop-Grumman, Cessna, Beechcraft, Airbus, Piper, HondaJet, Gulfstream, etc.  Preferably seek out internships in flight testing.
Finished with school?  Time to land your first job in flight test.  Attend your school's career fair and chat with the reps from the OEMs.  File applications for jobs on OEM careers sections.  Get your LinkedIn page set up.  It's now time to get references from your internship supervisors.  This first job is always a challenge as many OEMs are looking for experienced people and there are very few entry level roles.  You first job may not be in flight testing and will probably not be very glamorous.
Aerospace is a limited and challenging field to get into.  Work is also limited to certain areas of the nation, primarily Seattle, LA Area/SoCal high desert, Dallas, Wichita, Greensboro, or Savannah.  In addition, the business is quite fugitive, having many boom and bust cycles.  Layoffs happen very frequently and you may have to move from place to place to find work.  Starting salaries and benefits packages can be spartan.
Flight testing may or may not be as glamorous as you think it is.  Your primary duties will be generally doing test plans and reports for flight test operations, rigging aircraft of a/c systems with sensors, wiring and data collection boxes, possibly some 3-D CAD modelling.  Even the engineers on the aircraft are sitting at crew stations monitoring data collection.  After that you'll be doing data analyst activities, crunching numbers, etc.
A PPL or CPL is not necessary to be a flight test engineer.  Test pilot is another story altogether and is considerably more difficult.  There are civilian routes into this business but are extremely difficult to obtain.  Most test pilots get their education by becoming military pilots and attending either the USAFTPS at Edwards AFB or the USNTPS at NAS Pax River.  In addition, most of the test pilots at the major OEMs come from the military route and, IMO favor the military guys over non-military, particularly in flight testing high performance military aircraft.  Near as I know there are no requirements from the FAA to become a test pilot but the only private test pilot school in the nation is the National Test Pilot School in Muroc, CA.  They require applicants to possess an engineering degree, a CPL and at least 750 hours total time in fixed/rotor wing.  With an 11 month course to obtain a Masters of Science in Test Piloting costing an astounding $960,000.00, you will definitely need corporate sponsorship.  There are a few firms which will do this.  Burt Rutan's Scaled Composites in Muroc promotes test pilots from its engineering corps.  So does Cirrus in Duluth, MN.  But the competition to obtain these slots is fierce.  All in all, if you want to be a test pilot, go the military route, either in the Military Academies or a ROTC unit at your college.