How do I connect my Time Capsule to an Existing Network?

I've seen a few similar discussion threads and seem to suggest I can't do this without an Apple Router, but I just want to clarify:

I have a 3TB Time Capsule, I want to connect it to my existing Wireless Network so that I can set it up with Time Machine.

I have connected it to my current router (which is AT&T, not Apple) via an ethernet cable that's plugged in to a LAN port on the router, and the WAN port on the TC (I also tried this in a LAN port on TC but it didn't seem to make a difference).

I was able to find my TC on Airport Utlility, but when I try to select the Join Existing Network option it just keeps bringing me back to the Create New Network screen.

How do I fix this? Does it really need to be an Apple router? Am I just plugging the ethernet into the wrong port? Or should I just continue in the Create New Network screen and enter the same name and password as my existing network, or would that create a duplicate network?

Honestly I don't really care about extending the network or not, I just want to be able to wirelessly access the Time Capsule to use it with Time Machine.