This can be done in any Game Mode except for Time Trial.
In the Assist settings, be sure to turn everything off or to its strictest setting.
For Gear Shifting, choose Manual or Manual with Clutch (either works for the Achievement).
I recommend doing the race with a Modern Formula 1 Car as some Classic Formula 1 Cars won't change gears on Manual with Clutch on a controller.
Complete any race with these settings and the Achievement will unlock regardless of race result.
Note: Flashbacks are not considered Assists and can still be used.