You can get a watch knife from a catalog shop or the Internet. I really recommend the Victorinox since its blade is sharp but sturdy and can help with very tight covers.
Find a notch on the cover and start inserting the blade there. Make sure you keep the blade horizontal to avoid it to slip and scratch the watch. Increase the pressure until the back cover pops out.
Slide the blade to another side of the back to remove it completely.
Use the tweezers to move the battery support and remove the battery. Make sure you replace the battery with another of the same type.
After replacing the battery make sure the watch is working correctly. If it doesn't check that the battery sits properly and the polarity is correct.
If it still does not work you may need to clean the contacts. Sometimes the batteries can leak and corrode the contacts. Use a contact cleanser if the corrosion was too much. A dry cloth works too.
Put the cover back and snap it on with your thumb. Make sure that surface beneath the watch is clean. Otherwise you might scratch the glass while pressing the cover back on.