You can't plug a ethernet cable into a computer monitor and expect to get a signal. Most obviously because the monitors won't even have a ethernet port on them, but that is because a network plug isn't for sending video signals. Sure, you can view video that was received through an ethernet connection, but that is after the device/computer processes the network data and creates the video output. A computer monitor is not going to have any of the "processing" capabilities built in to do that. In other words, the data carried on the ethernet cable is not specifically a video feed like a HDMI or VGA connection carries.
If you are talking about a Smart TV, then yes you should be able to plug an ethernet cable into the device and then use the built in browser to view the camera feed. That is because the smart TV/browser handles the "processing" needed to view the camera feed.
As already noted, there is no such thing as an ethernet splitter, you need to use a network switch to connect two or more devices to a single network cable.
Finally, most computers can support more than one monitor. It is very likely you can simply plug in a second computer monitor into the existing PC and use it to view the camera feed while still using the original monitor exactly how you are today. Even if the computer doesn't support two monitors "out of the box" it is likely there is a PCIe slot that you could add a cheap graphics card (like sub $50) to which would give you a second monitor output. However I would be very surprised if your existing computer didn't support two or more monitors already.
