This is technically story based. Not really missable, as you can talk to the housekeeper at any time.
After progressing so far into the story line;
return to your room. Here you must speak to the housekeeper, who will give a brief tutorial on moving rooms, telling you that you deserve some RnR. You can then select the 'Move Room' option and select Private Quarters, once you load into there, the achievement will pop!