Rare Replay

In order to be able to obtain platinum medals you must first get 2 gold medals in all the main Carrier missions as well as 1 Gold on every Bonus mission.

Once that's done you'll unlock the time trials for the carrier missions, get gold in all of these and you will be able to start earning platinum medals.

Make sure to end the level immediately as the time will be saved for the time trials

If you have gotten gold in every level but the secret offworld missions aren't showing make sure to check if you have found all CP's you'll know if there is one left as the level will have a green ring in the mission select.

As for the medal itself, I found the very first level, Simian Acres to be the easiest.
The target time for this mission is 14 secs.

As soon as the level starts just drive through all the necessary buildings, aslong as you destroy all buildings without stopping or turning around you'll get it easily.

This is the easiest mission to get platinum.
There is some kind of glitch/workaround in the mission Skerries.
Just follow the video.

Just a tip for those, like me, that didn't know better. When you are playing through the regular, main campaign, make sure to end levels once you clear the path instead of just letting time run out. You can end levels by getting out of your vehicle and getting into the Blast Corps semi-truck. This will record the time you ended the level at for later when you unlock Time Trial mode.

I didn't know this and just let time run out while I zoomed around destroying buildings and collecting RDUs. Then when I unlocked Time Trial mode I had to redo almost every level. If you end the level, you should have golds for most of the levels and won't have to redo really annoying levels like Ebony Coast, Diamond Sands, and Oyster Harbor.

Also, remember that in Time Trial mode the carrier doesn't actually need to make it all the way to the end, just all the buildings need to be cleared. That means you can do things like, not fill in all the holes in the road on Oyster Harbor. As long as the buildings are destroyed before the carrier blows up, you'll beat the level. I had to redo Oyster Harbor and only filled in the first 3 holes to buy me some extra time.

Finally, if you struggle with Simian Acres when you are trying to get your platinum, try some other levels. Other people have recommended some of the race ones, the J-Bomb level, and Skyfall as alternatives.

I personally did it on Skyfall after a couple of attempts after watching a video. On Skyfall you want to hit one of the silos on the end without boosting and then use all your boost to run through the other three in under four seconds.

Good luck!

By design this will basically be the last achievement you get in this game. You will need 2 Gold medals on all 21 main missions (1 for completing the level and 100%ing it's objectives, as well as one near the end of the game for speed on completing the level) as well as Gold medal on all 36 side missions.

SPOILERS!!! Be aware this guide will cover some mission related secrets and spoilers as those are necessary in order to earn this achievement.

Some tips along the way: Always end the mission by jumping into the Yellow Truck and always finish it as fast as possible (as you will have to go back later on and improve some times if you don't). Also to get a Mario Kart style speed boost at the beginning of each level accelerate when you hear the last beep as the light turns green.

Below is the list of missions as well as the time needed to get a Gold on each one. You have to get a Gold on all 57 missions before you have the ability to earn ANY Platinum medals! I found the easiest level to get platinum on for me was Beeton Tracks.
Legend:
G= Gold, P= Platinum

Easy Levels:
1 Simian Acres: G :25 P :14
2 Blackridge Works: G :40 P :23
3 Carrick Point: G 1:10 P :36
4 Having District: G 1:10 P :28
5 Argent Towers: G :50 P :28

Medium Levels:
6 Echo Marches: G 2:55 P :45
7 Beeton Tracks: G 1:35 P 1:15
8 Tempest City: G :52 P :27
9 Outland Farm: G 1:45 P 1:00
10 Ebony Coast: G 1:45 P 1:12
11 Ironstone Mine: G 1:30 P :54
12 Shuttle Gully: G 1:25 P :44
13 Cromlech Court: G :36 P :22

Hard Levels:
14 Obsidian Mile: G 1:45 P :55
15 Crystal Rift: G 2:05 P :57
16 Glory Crossing: G 1:10 P :40
17 Ember Hamlet: G 1:30 P :40
18 Angel City: G 2:50 P 1:35
19 Diamond Sands: (Very hard) G 3:40 P 2:35
20 Oyster Harbor: (Very hard) G 6:00 P 2:55
21 Shuttle Clear: G 1:10 P :54

Side Missions:
22 Sideswipe: G :40 P :12
23 J-Bomb: G 1:20 P :30
24 Backlash: G :22 P :11
25 Sleek Streets: G :50 P :40
26 Kipling Plant: G :20 P :12
27 Cooter Creek: G :35 P :25
28 Skerries: G 2:00 P 1:35
29 Twilight Foundry: G :35 P :29
30 Orion Plaza: G 1:12 P :36
31 Marine Quarter: G :40 P :36
32 Jade Plateau: G 2:12 P 1:27
33 Salvage Wharf: G 1:00 P :44
34 Thunderfist: G :36 P :18
35 Skyfall: G :12 P :04
36 Morgan Hall: G 1:24 P :42
37 Corvine Bluff: G 1:06 P :55
38 Bison Ridge: G 1:36 P 1:20
39 Cobalt Quarry: G :40 P :36
40 Glander's Ranch: G 1:45 P 1:33
41 Silver Junction: G 1:00 P :30
42 Moraine Chase: G 1:10 P 1:01
43 Mica Park: G :50 P :30
44 Falchion Field: G :30 P :21
45 Dark Heartland: G 1:30 P :44
46 Gibbon's Gate: G 2:00 P 1:15
47 Geode Square: G :40 P :20
48 Lizard Island: G 2:30 P 1:30
49 Saline Watch: G 1:40 P :45
50 Dagger Pass: G 2:00 P :50
51 Magma Peak: G 2:00 P 1:00
52 Baboon Catacomb: G 1:10 P :45
53 Moon: G 4:00 P 2:20
54 Mercury: G 1:20 P 1:10
55 Venus: G 3:20 P 2:15
56 Mars: G 5:00 P 1:50
57 Neptune: G 1:40 P 1:28