To earn this achievement, you must win five rounds in arcade mode without taking a SINGLE PIXEL of damage. That means if you get hit by a special move and take "chip damage," your perfect is ruined.
I earned this achievement using the M.Bison "Psycho Crusher" method mentioned earlier in the section "Sheng Long is in Another Castle." I was playing on easy difficulty of HD REMIX mode.
Be careful though. I found that once my opponent was one Psycho Crusher away from death, they would finally counter the move. I remedied this by using a normal attack instead of a finishing Psycho Crusher. This will not work every time, but it should work enough to get you the achievement.