First make sure your existing wireless network is created by an Apple AirPort Base Station. Make sure the Extreme is well within its wireless range and not approaching its limits. Then, configure the Extreme to "extend" the network as explained in the above article. It usually helps to start with a "hard reset" to eliminate confusion with existing connection methods.
Once configured, its LAN ports should serve as additional wired ports for your non-wireless computer.
"Joining" a wireless network will leave its LAN ports disabled, so that won't be an option. It must "extend" the network to be able to do what you intend.