Thank you for using Apple Support Communities. It's my understanding you'd like some assistance with cleaning your new iPhone case. I have a similar case on my iPhone, it's important to me that my iPhone and all accessories stay clean. I'm happy to help.
Please review the information below:
Cleaning your Apple productsiPhone Smart Battery Case
To clean your Smart Battery Case, remove your iPhone from the case. Use a soft, slightly damp, lint-free cloth to wipe the outside of the case. Don't use window cleaners, household cleaners, aerosol sprays, solvents, ammonia, abrasives, or cleaners containing hydrogen peroxide to clean the case.
When cleaning the internal Lightning connector, use a soft, dry, lint-free cloth, and don’t use liquids or cleaning products
Have a wonderful day!