you dont need to transfer anything, the details are held on the router. think of the router as a small PC on its own.
If you connect the PC to the router by ncable then simply plug it in.
If you connect by wireless then just enter the relevant details onto the new pc as per the settings on the router
ie SSID, Type of Encryption, encryption key.
If you have set up MAC filtering on the router then you will need to add the MAC address of the wirelss card of the new PC to the access control list.