How would I go about using my warranty for an Apple phone case?

All Apple Accessories are warrantied for a year. If the case is less than a year old and the issue is not cosmetic damage, you take it to your Apple Store and show it to them. If they agree it's a manufacturing defect they'll replace it. If it's older than a year you don't have any warranty available.

Do you mean a relatively new case has an issue. Call Apple or make an appointment at an Apple Retail Store location if one is nearby.