Yes, the ASUS needs an LAN port to LAN port attachment to the Linksys router (through the Ethernet cable that I assume is in your walls to the desired location). Then configure the ASUS as an AP (access point). Consult your manual for details, but the setting is in Advanced settings -> Administration -> Operation mode. Select AP Mode and click Apply.
You will need to give the ASUS a static address in BOTH the ASUS and the Linksys that is in the network range but outside the Linksys DHCP range (DHCP will be off in the ASUS). So for example if the Linksys gateway address is 192.168.1.1, make the ASUS 192.168.1.2 and set the Linksys DHCP service to use 192.168.1.3 and above.
If the Linksys does not use its wireless then just set your WPA2/AES security with a password in the ASUS (if the Linksys does have the wireless radio on, use all the same settings and SSID only change the radio channel to prevent interference).