Is it fine to use a battery case for iPhones that is a unknown brand that uses a Apple charger to charge the case but the battery cas wit self has no sign of mfi approval?

Hi

When buying electronic accessories for your iPhone, ensure that they are either genuine Apple products or that they have been certified as Made for iPhone under the MFi program. Certified items come with the "Made for iPhone" logo on them and/or on their packaging.

Genuine Apple Lightning to USB cables have certain text and a serial number printed on them.

Apple Smart Battery Cases, for example, are available for certain models, including from the Apple Online Store:

http://www.apple.com/shop/iphone/iphone-accessories/cases-protection

Using uncertified or counterfeit accessories may damage your iPhone and/or adversely affect operation. Apple's warranty does not cover damage caused by use with a third party product that does not meet Apple's specifications.

Apple also recommends using the Apple USB Power Adapter that came with your iPhone (or another compatible Apple power adapter).

More information:

About iPhone, iPad, and iPod accessories - Apple Support

Identify counterfeit or uncertified Lightning connector accessories - Apple Support

If you could provide some specific information about the case you bought (Brand, Capacity, etc...) it would greatly improve the chance of having a better answer!