Hello my name is ***** ***** I can help you with your question.
About your Panasonic HC-V10:
When you plug in the camera with the USB cable, the status indicator will flash red for approximately 2 seconds indicating that charging has begun. It will turn off when the charging is completed. If it's not flashing red that means the battery is fully charged!
Here is a link the full User Manual, and turn read page 13 more more details.
If there is anything else I can help you with, please let me know.
Yes. The battery is the most likely problem. When you do order the replacement, be sure that it is a genuine Panasonic battery. That will give you the best result.