Convert BNC to Ethernet to Coax/HDMI
Edit: OP has added that the Cat5 runs are connected via an Ethernet switch. Therefore, original answer (left below) is incorrect.

Because the cabling is attached to a switch, you cannot connect analog video to the switch. There are a few options:


Ignore the ethernet and separately run your own cable (either coax or Cat5).
Depatch the ethernet from the switch and couple the ends together. Then you can use baluns (make very sure that the cable is no longer connected to the switch).
Buy a composite video to IP video converter. AFAIK, these are not very cheap. But they allow you to connect the analog camera into the network.
Ditch the existing cameras and buy IP cameras that can connect directly to your network. Low-quality IP cameras are very inexpensive, and are very easy to manage once connected to the network.


Once the video is on the network, you would need a small computer to view the stream on the TV.



Original answer

You can buy BNC-RJ45 baluns; they're all over the web. In fact, you can buy a nice little unit that runs four BNC lines into a single RJ45 connector. You should expect a little bit more loss than coax, but it's hard to say how much the image quality will degrade.

If passive baluns don't cut it, there are active converters that use proper differential signalling and have some amplification/pre-emphasis. As it's hard to know which you need, I'd suggest buying passive baluns, see if they work, and if not get some active converters.

I should point out that this solution will only work if you are sure that it is a single continuous Cat5 run. If the Cat5 is run to a switch, this will definitely not work and may cause the switch to be unhappy. If you want to run video over a network, you are better of just buying IP cameras that can run video directly over the network.