Although this achievement sounds like a pain in the ass (and looks it by the maps perspective) it's not really all that hard. There are a total of 315 roads on the map which require you drive down all the way Untill the greyed out line on your map becomes white in Colour.
You are also required to drive down the orange faded shortcut lines turns bright orange for this achievement also!
Driving around looking for boards, barns, speed traps bucket list events and just normal races should unlock most of these roads anyway just by working towards other achievements :)
Another thing that you should also consider is driving back and forth along each road twice as you can miss a little section of the road and it can become hard to find as the white will make it look like there is no grey lines.
Note: if you use
while in the map, you can filter it so that all the icons are turned off making it easier so you can see the roads have and don't. (Credit Goes To: ScuzzyBunny)
Note 2: a tip by Hanrec2 pointed out that sometimes roads may (in their case highway) did not join smoothly with the other roads which took going back and driving carefully along them to get them joining and unlock the achievement.