OK, thanks.
A normal home network configuration will go modem > router > computers, just as your network is setup now. The AirPort Extreme is also a router. There are sometimes reasons to have two routers in a home network, but things are much simpler if that can be avoided.
My opinion here....It really would not make good sense to further complicate your network to go modem > router Extreme > router Linksys > computers.
It would be much simpler if you consider pulling the old Linksys and putting the Extreme in its place. The Extreme has 3 LAN ports which could handle 3 computers. If you need more computer hookups, an inexpensive ethernet switch (not a router) $15-20 or so, could be plugged into one of the LAN ports to give you even more ethernet ports.
If you are intent on using the Linksys, it would logically come after the Extreme. So, you would plug the linksys into one of the LAN ports on the Extreme and then connect the 4 computers to the Linksys. The Linksys would need to be reconfigured to be in "bridge" mode to work correctly. I haven't configured a Linksys in years, but I assume that this is still done by accessing a configuration page on the web.
One other thought...if the Linksys is 4-5 years old as you indicate, it would not handle the faster ethernet speeds that are available today. That would be another argument for using a simple "switch" with the Extreme to give you all the connections that you need for your computers.
Hope this helps. You may get some different opinions on this, so take a little time to decide how you want to proceed.