The most common ClearOS installation has just one NIC (network interface card) This is set EXTERNAL but really it's just LAN with a gateway (your Linksys). To use this setup simply turn off DHCP in your Linksys and set it up in ClearOS. Make sure ClearOS hands out the Linksys IP for the fateway like the Linksys normally does.

You should also make sure ClearOS has good DNS settings and then make DHCP hand out ClearOS as the DNS to the PCs. I'm assuming you set a static IP address for ClearOS by this stage.

You will easily get the above working since it's not much different from what you already have working.

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HOWEVER there is another way to use ClearOS as the Router itself, which is what you have been failing to do. This is what it was for in the days when it was Clark Connect.
You need two NICs. One is EXTERNAL connected just to the Linksys and the only thing plugged in the Linksys. The other is LAN where your PCs are plugged. You will have to turn off WiFi on your Linksys too.

Modem Mode on my Virgin router handles the Virgin login for me and hands my actual router the public IP. In your case you may need to set PPoE and have your ISP username and password in the settings in ClearOS EXTERNAL NIC. This is why it gets difficult. ClearOS then becomes your router and the Linksys is nothing but a modem. Set ClearOS DHCP accordingly so it's the DNS and gateway.

I suggest you go with the first method to begin with.