CAT5 is typically Unshielded Twisted Pair (UTP) and there are four balanced signals. They do not get grounded. Unbalancing these signals (like trying to ground one of them) would result in a non-working system.
There is a "Shielded Twisted Pair" that the shield could be grounded, but this is really expensive and most likely not needed.
You might want to consider grounding the housing for the camera.
Actually the cable I bought is shielded (see below). Since the cable is shielded is it necessary to ground the housing...The housing is under an eave on the first floor of my house...
80ft Cat5e Outdoor Waterproof Ethernet Cable Direct Burial 80 ft Shielded
Shielded RJ45 Connectors
Can be buried underground or used outdoors
Waterproof (Outdoor Rated)