Reach Affinity level 25 with any 4 car manufacturers.
As you complete races you will earn XP based on the length of the race that is used to level up both your driver level and your affinity level for the car manufacturer. This XP is not affected by assists or Drivatar difficulty (those both affect how much money you make, not XP). The formula for leveling up is (Level * 20,000) - 15,000 = XP needed for Level. Using this formula you can see that level one is (1 * 20,000) - 15,000 = 5,000. Level 25 is (25 * 20,000) - 15,000 = 485,000. When purchasing cars for the various Championship Series, try to focus on as few manufacturers as possible. Ford and Ferrari are common in many of the series, followed by Audi and Chevrolet. It is possible to get one or possibly even two manufacturers to level 25 in the natural progression of the various Championship Series. However, it is also possible to complete all of the Championship Series and hit level 150 driver level without a single manufacturer hitting level 25 affinity.
While you can always race online or in bonus races as well, to get the most impact out of your time you have two choices to efficiently raise the affinity level.
Active driving method:
From the main Forza screen, choose Multiplayer, press to create a private match and to confirm. Once the vehicle selection screen opens, choose a car from the manufacturer you are going to focus on. If you have multiple cars from the manufacturer, choose the fastest one. At this point you will be in the lobby, so press to change the options to match these choices:
Game Type: Circuit Race
Track: Indianapolis Oval
Max Players: 2
Max Drivatars: 1
Drivatar Difficulty: Set this at the level you normally race at, or at a level where the AI doesn't fall too far behind. You need to come in first place, but not by multiple laps. If you set it too far down such that you are lapping the AI multiple times, the developer has controls in place that can give you no credit for the game and it will waste your time.
Laps: 50 (depending on the top speed of your vehicle, 50 laps should take 40-80 minutes, so if you have less time, choose fewer laps)
Collision Mode: Always Off
Car Class: Any, or set to match what you are driving
XP rewarded: 51,000
From the main Forza screen, choose Multiplayer, press to create a private match and to confirm. Once the vehicle selection screen opens, choose a car that is rated B or below. At this point you will be in the lobby, so press to change the options to match these choices:
Game Type: Drag
Track: 2-player 1-mile drag
Max Players: 2
Max Drivatars: 1
Drivatar Difficulty: New Racer
Car Class: Any
When you go back to the lobby, all the Drivatars will have a class of car that matches yours. Press to go back into the settings and change your car to the highest rated vehicle you have from the manufacturer you are going to focus on. You will end up back in the lobby with your fast car racing against the lower rated vehicles the Drivatar is using, making the drag race basically an automatic win. Now have the right trigger of your controller compressed (use a rubber band or even the cord to your headset) and the car will automatically accelerate down the track, winning every time. It will take approximately 10-15 minutes to complete the 10 heats, at which point you will be back in the lobby. You can speed this up by pressing between heats, but if you are going to be around for that you are likely better off using the active method detailed above. Press to restart the process.
XP rewarded: 10,000
This achievement will unlock when you get your fourth manufacturer up to a level 25 affinity. Note that you can boost this using the AFK method at the same time you are getting multiplayer medals if you have one other human racer in the room with you, just switch it to an 8-car 1-mile drag with 6 Drivatars.
To view your affinity levels you can go into the Forza Profile from the main menu and choose Affinity in the menu at the top. This will show you every manufacturer you have driven a vehicle from, your current affinity for each one displayed to the right, and a progress bar showing how long until the next affinity level. If you scroll down through the list, more detail is provided in a stats box to the far right.