I was able to get this on the third grand prix - Racing Elite using the Carbon.
Winning every race in the grand prix isn't needed but you will need to win overall, so I recommend restarting the first race if you don't win to give yourself a better chance if you don't do very well in the later races.
A few tips:
-Start holding RT as soon as you see the 1 on the countdown, this will give you a boost launch every time
-Be sure to make use of slipstreaming (driving behind another vehicle) to increase your speed and build boost
-You can 'latch on' to other vehicles and be able to go as fast as they're going by driving into the side of them or getting pushed from behind, but this may cause you to crash or even spin if you're unlucky
-Practise the tracks in time trial or on a lower difficulty if you are struggling
-Have a bar of boost left over at the end of the race in case you need the extra boost to get the win
-The AI can seem very broken with rubberbanding near the beginning of a race but you will most likely catch up to them near the end
This isn't my video but it shows the general routes through each track, full credit to Marius Urucu for the video