Your screen or your proximity sensor could be defective. If not the proximity sensor could of gone bad from a drop. They are both replaceable, you can use your voice memo to test your proximity sensor by recording something and playing it back. Just put your finger over the proximity sensor and see if it blanks out or if they screen stays black.
I would recommend you take the device to an experienced shop. The iPhone 5S is very touching with the 64 bit operating system and your phone could be destroyed with one wrong move. I would also try connecting your device too itunes and updating your software. Apple has been having a lot of issues with their software. Hope this helps.