This is best accomplished on the level that's called Facility and it must be done on the 007 Classic Difficulty in order for this achievement to unlock.
Unlike the I Am INVINCIBLE! achievement, you can get hit all you want. What you cannot do is stop to get any kind of body armor that's found lying around in the level. To make this easier for you, the recommended weapon of choice is the P99 Silencer. Make sure you take out your opponents with accurate head shots. In some situations, there will be two guards next to each other. Kill one with the first head shot, then unload your bullets on the other guard until he's down. In short, just go through the level as normal as you would without collecting any body armor. It doesn't matter if you complete the extra objectives; as long as you complete the primary objectives without any body armor attached, you're good to go.
To see how this achievement is earned, please check out the video below. Credit goes to HDSmity for creating this video and having it shown to the public. Although this covers the I Am INVINCIBLE! achievement, this also covers the Dressed To Kill achievement, which is the primary one we are going for. It may take a few tries, but you'll get there without much of a problem if done correctly. Good luck!