To install the network printer you need to follow a few steps- first, do you have the driver for the printer? If not, go to HP's website and download the latest available, be careful about x86/x64, save it somewhere on the hard drive. If its zipped, extract it to a folder on the hard drive.
Now, you need to make sure the computers on the printer can actually see the printer. For this, first connect the printer via cat5 cable to the router (any LAN port) and print out the config page using the buttons on the front of the printer. If your router is assigning addresses dynamically, the printer would show up a dynamically assigned ip address. If you get nothing at this step, either make sure dhcp is enabled on the router or assign a private static ip to the printer. Let us know if you need to do this.
Now on any PC thats connected (wired/wirelessly) to the same router, go into "devices and printers" page and click "Add a printer" at the top. Let it detect the network printer, if it successfully detects it means the network config is correct. Follow the onscreen prompts and when it asks for the driver, point it to the folder where you extracted the driver. Another way of going about this is to install the driver right at the beginning, the printer installer will automatically find the installed driver.