When I try to connect the Amazon Fire TV device to my TV using a HDMI cable, all I can see is a black image on the TV....

There could be a couple reasons for the Internet connectivity problems. Maybe the Ethernet cable is not correctly connected, or the router is not providing Internet access. Maybe it's also a hardware problem, in which the Ethernet input port may be damaged. For more information, visit Amazon Fire TV Troubleshooting

The device's remote might not be working for many reasons. First, you need to make sure the batteries have charge. The next probable cause might be that the spring that touches the battery's negative terminal is stuck. If your remote control still doesn't work after that, maybe there is something wrong with the contacts between the button panel and the motherboard, or the motherboard itself has some kind of damage. Check out this troubleshooting page: Fire TV Device Troubleshooting You'll probably find the solution there!