What would happen if I connected my PC to my set top box with an ethernet cable? My tv only has 1 hdmi port.
The Ignite TV set-top box is just a special media streamer, similar to an Apple TV or Roku box. It gets its network connectivity using either Ethernet or Wi-Fi and outputs video through the HDMI port. It can't work any other way. You cannot output video through the Ethernet port to your computer nor can you use the Ethernet port on the set-top box to provide network connectivity for your PC.
Does your TV have any other inputs other than the single HDMI input? If so, you might be able to connect the set-top box using an appropriate HDMI to AV or HDMI to Component converter. You can also connect multiple HDMI devices to your TV through an HDMI switch.