This guide was a requested guide. If there is an achievement guide you'd like to see, send me a message on Xbox Live. Take a look at my achievements list to see what I have unlocked.
NOTE: There have been several reports of this achievement not unlocking, I'll go over that shortly.
To start off, here are the following cars I have used for each mission type.
Perf - Lamborghini Huracan LP 610-4 tuned to 1373.
Street - Nissan Skyline GT-R (R34) tuned to 1341.
Dirt - Nissan Skyline GT-R (R34) tuned to 1271.
Raid - Nissan Skyline GT-R (R34) tuned to 1094.
Circuit - Lamborghini Huracan LP 610-4 tuned to 849. (These races were easy and not many, so I didn't put much effort into this spec)
The easiest way to level up your cars is to do the skill mission "Scramble Skill 12" located at the Chicago Airport. Come at the start point from the North-West. (This seems to be the easiest at getting Platinum medals)
Now that you have your cars to spec, go into the missions and go about them each as fast as you can. For raids, if going after them in a certain order doesn't work, go after them another way. Sometimes, the best route isn't the one you might think first. Remember, you can afford to miss a stack here and there if it means you keep your speed.
Play each mission several times and look for short-cuts or places to go off-road if it'd be faster. If you screw up, start again.
Also, it helped me to go for platinums starting from the missions I got a gold on, then silver, then bronze. (Bronze were first missions or harder ones, so I saved them for last)
Remember! If you pull up your map and click the left thumb-stick (Open Driver ID) you can scroll to statistics and go under "Leaderboards" to see how many platinum medals you have unlocked. Click on a region then hit missions to see your medal for each mission.
If this helped at all, please let me know!
EDIT: I forgot about the glitch. If the achievement does not pop up when you have finished everything, go through the missions again or go through Skills and hopefully it will unlock. If that doesn't work, then I am sorry. I have no solution other than trying skills or doing the missions again.