This is a very easy method to complete the Modern Warfare 2 campaign on veteran difficulty and obtain the achievements. Just follow the steps below.

1) Make sure you have more than one storage unit such as cloud storage and a HDD.
2) Select the mission you want to achieve veteran on.
3) Start the mission on Recruit difficulty.
4) Play the mission through to the last checkpoint.
5) Save and quit.
6) Change the storage device to the device you were not using before.
7) Restart the level on veteran.
8) Save and quit.
9) Change back to original storage device.
10) Continue the campaign.
11) You will spawn at the last checkpoint of the mission being played through.
12) Simply resume your game and complete the last checkpoint on veteran difficulty.
13) You will receive the stats and achievement.
14) Repeat this for every mission to rack up the achievements.

Here is a video tutorial: