Camera profiles are stored to the following locations:

Mac
/Users/<user name>/Library/Application Support/Adobe/Camera Raw/Camera Profiles/

PC  **You must set up your Folder Options (File Explorer Options in Win 10) to View Hidden Folders to be able to navigate to these paths.

• Windows Vista, Windows 7, Windows 8, and Windows 10 - [root drive, typically
C:]\Users\[username]\AppData\Roaming\Adobe\CameraRaw\CameraProfiles
• XP - [root drive, typically C:]\Documents and Settings\[username]\Application
Data\Roaming\Adobe\CameraRaw\CameraProfiles

You can also use the DNG Profile Manager which will help you to make a much more automated process.  To delete a profile, select it in DNG ProfileManager and click ‘File / Delete’. This moves your profile to the Trash (Mac) or Recycle Bin (PC).  You should be able to download a current copy of the DNG Profile Manager from the following link:

https://www.xrite.com/service-support/product-support/calibration-solutions/colorchecker-passport-photo