I believe that would qualify as class then race then background, or class then background then race. Ability scores come before class, but class comes before the other two things the poll is concerned about.You're the missing pre-WotC character creation option. Roll stats and see what classes the character qualifies for, then go from there.
My normal approach is to figure out a concept, then figure out which mechanics to use. Sometimes it's a race, sometimes a class, sometimes a specific game mechanic or spell, but I use this as my starting point. With 5E, I've DMed about half the time, so I don't get many chances to try out different characters. Because of this, I usually pick a subclass to build off of, such as my current Nature Cleric, Fiendish Warlock, and upcoming Bladesinger.