CQB = Close Quarters Battle (if you wanted to know)
Main key points:
1. Bullets will penetrate trough thin, weak material such as wood, plaster, and sheet metal. The cargo ship mockup is composed of mostly plywood.
2. Firing from the hip is okay, although it will decrease your accuracy. This means that you DON'T need to aim (with the Left Trigger) to fire your gun all the time.
3. Study the cargo ship mockup as much as possible. Before heading up the latter, you can go in where the test ends to get a good view where silhouette targets are going to pop up.
4. You DON'T need a perfect line of sight. If you know where the silhouette targets are, then you should know at what angle you could hit 2 of the silhouette targets at the same time with 1 bullet.
5. Flashbangs take a little bit more time to throw than stun and smoke grenades. You can "prep" a flashbang (by holding the Left Bumper) after hitting the target that is at the bottom of the stairs.
6. DON'T be in the same area where the flashbang goes off. You'll be affected by it and will prevent you from being able to sprint or even see what you are doing for a short period.
7. Sprint whenever possible (especially when heading from one position to another), it will knock off some time.
8. Accuracy is very important if you aren't very fast. Being accurate will knock off 3 seconds from your time when you cross the finish (i.e. 22.99 seconds becomes 19.99 seconds, which would unlock the achievement).
9. Switching to your pistol (Y button) is a lot faster than reloading your primary gun (MP5).
10. And finally... DON'T use your knife! It doesn't help you any in the CQB test. Well... unless you want to smash a few TVs along the way. If you are within the knifing range of a silhouette target, then you should just fire your gun from the hip (i.e. not aiming down the sight with the Left Trigger).
Update 7 April 2019:
I have my own video (no commentary) as I recently did this achievement in Call of Duty: Modern Warfare Remastered
on Xbox One.
A friend of mine, Speak79
, has done some editing and posted it via his YouTube channel (with permission from me). He slows it down to 50% so that you can see at just how accurate I get with this run.