Select Single Player from the main menu then World Challenge Arcade mode, there you'll find the "Platinum Plus Challenge". All you need to do is complete every event in this chapter (you cannot change the difficulty). The achievement will unlock at the end of the last race once you've earned medals in all of the events.
Event 1: Street Race - Ultimate Aero TT (difficulty 10/10 - manual gears)
This is the toughest of the challenges, mainly because the car is very powerful, takes longer to brake and taking place on a narrow snowy track! Manual gears is a must here due to the rear spinning up so easily in auto. Ensure you use a higher gear than usual to prevent losing the back end. Most tight corners should be taken in 3rd. As with all platinum races, try to gain 1st place during the 1st lap and then I found it's easier to keep the rest of the pack at bay. Brake very early and avoid drifting! The second half of the track where there are many twistys is a great place to overtake.
Frustrating, but after learning the track and car handling you should be OK.
Event 2: Superstar - Cobra 427 S/C (difficulty 9/10) (see YouTube video below)
Time is the main problem here I found. Try to aim for 25 stars per lap but don't get caught trying to snake down the main straights, these should be flat-out sections. Dab the handbrake to drift the car and listen for the 5 stars, corners #2, #4 and after the tunnel section should be 5* corners. It's tight but try to be smooth and maintain forward motion as much as possible. Be aware of staying on the power too long will step the rear of the car out too much losing you valuable speed. I managed 100 stars with 1 sec remaining on the clock.
Event 3: Timed Run - NSX-R (difficulty 7/10)
The easiest of the challenges which shouldn't be too much of a problem once you're familiar with the course and car handling. Again, braking early and hitting each apex will provide low lap times. Watch for the standing water which can upset the car. Avoid drifting and learn the parts of the track which can be taken flat out. Braking early for the penultimate corner will enable you to sling-shot out leading to an advantage down the main straight. Aim for a 54 second 1st lap, if lower just restart.
Event 4: Speed Challenge - Turbine Superbike (difficulty: 8/10 - manual gears) (See YouTube video below)
Use manual gears on the jet bike - don't worry it only has 1st and 2nd! Ensure you're straightened up after the slow corners (brake early down to 40MPH) to then aim reach at least 100 MPH through those checkpoints (must stay in 1st gear). As you're passing these cameras pull a small wheeley for a few extra MPH. Hit the speed camera at the end of the main straight at full speed (234 MPH) resulting in a crash. You're aiming for 533 MPH total after lap 1 and 1066 MPH after lap 2, if short hit the restart.
Event 5: Street Race - Corvette Stingray (difficulty 9/10)
As this car is slower than the others you'll need to perfect the cornering, ensuring early braking in a straight line and always avoid drifting. Learn these braking points, especially after the long straight. Don't be afraid to use the other cars to help you around the corners too! Bends 3 and 4 (right-hand kinks) can be taken at full speed and usually you can gain a place or two here. It's then possible to grab 1st place by the following sharp left corner. Once you're in 1st place continue to drive smoothly and if required block the AI on the main straight - play nasty!
As with all Platinum challenges, practise makes perfect. Good luck!