Q: Set up Time Capsule as Router behind Cable Router?

You first need to figure out how to login to the configuration of your Cable model/router.  Then, research how to put that device into "bridged" mode.  Doing so will turn your ISP's modem/router into a simple bridge that simply passes packets along to the WAN port on your TimeCapsule, making it your router instead.

Most gateways supplied nowadays cannot be bridged by end users. You can buy your own but tell us who your ISP is before you rush out and buy one.. it must be on the compatible list for the ISP and don't buy any of the single port surfboard series from ARRIS (ex motorola).

I would like to set up my Time Capsule (just purchased) to serve as a wireless router and backup drive

There is no need for the TC to be in router mode.. it will function just fine in bridge with wireless functional and backup functional. You will need to run ethernet to the new location of the TC though.. you cannot wireless connect the TC to the existing cable router. If installing ethernet is not possible sometimes EOP adapters (powerline adapters or homeplug) will work ok but there is no guarantee. Just remember it cannot use wireless.

I have FiOS for TV and internet.  I know that I can log on to the router, but haven't done it in a while.  Will look tonight to see if I can set it up in bridge mode.

I can't tell whether you mean to say that you want to set up the FIOS gateway in Bridge Mode, or set up the Time Capsule in Bridge Mode.

I doubt that it is even possible to set up the FIOS gateway in Bridge Mode, and even if you could, you would likely lose some of the FIOS services.

Connect an Ethernet cable from one of the LAN <--> ports on the FIOS gateway to the WAN "O" port on the TC

Perform a Hard Reset on the TC by powering it on for a few minutes....if it is not already powered on....and then holding in the reset button on the back of the TC for 7 to 8 seconds, then releasing and allowing a full minute for the TC to restart.

Then, use Apple's configuration "wizard" to set up the TC. If you not familiar with the wizard, you get it started by clicking on the WiFi icon at the top of your Mac's screen. click directly on Time Capsule under the heading of New AirPort Base Station.  The example below shows an AirPort Express as the new device to be set up, but you will see Time Capsule on your screen.

That will get the setup "wizard" up and running. It will take a few seconds to suggest the correct setup and a window like the one below will appear, except that a picture of the TC will appear.

Network Name......Type in a name that you want to call the TC  wireless network.  If you want to "extend" the FIOS wireless network, type in the exact same name of the FIOS wireless network. Otherwise, type in a different name for the TC network

Base Station......Type in a device name for the TC

Password......Type in a password that will be used for both the wireless network and base station. If you elected to "extend" the FIOS network above, then you would type in the same exact wireless password that the FIOS gateway uses. Otherwise, type in a different password

Verify.....Type in the same password again to confirm

Click Next

The "wizard" will set up everything for you automatically and apply the correct settings to the TC, so you don't need to worry about jargon like "Bridge Mode".

When you see the message of Setup Complete on the screen, click Done....and you are done.  Now you can connect to the TC wireless network that you named and enter the password if required.