After completing a level, there's a prompt to "edit the reply" by pressing X. Do so, then press X again and the game runs through the entire replay and records it. The achievement unlocks immediately afterwards.
Instead of pressing X a second time, you can also hold A to record only a segment of the clip.
In either case, the replay is recorded and gets uploaded to your console activity feed and the achievement unlocks. You are now "a cog in an advertisement campaign."