If you're having a problem powering on your camera, first check to see that the battery is fully charged and that you are able to power the camera on using the power button on the inside of the camera. This is done by opening the back door of the camera and pressing the small square button that is between the microSD card slot and the battery.
If your camera still will not power on, verify that the battery is fully charged. It should take between 4 and 5 hours to fully charge your camera's battery. You may need to try using a different USB port on the computer and/or a different mini-USB cable.