First thing.. the Asus wireless router will not work as a wireless extender to apple router.
Apple use some proprietary settings.
You do need to plug the asus into the Apple Time Capsule.. (Time Machine is software on your computer, not hardware).. by ethernet and run it in WAP mode.. if that is missing use WAN Bypass which I have explained just recently in another thread which I will give you the reference to.
Add spare Linksys WiFi router as another access point
If the Asus is a recent one WAP mode should be available.. and just google it.. asus RT-AC66U WAP mode and see what comes up. (replace with your actual asus model, which would help to know). Instead of ethernet you can at times use EOP (homeplug) adapters instead.. however these work or don't work depending on your house wiring.
Then you setup roaming .. both devices use the same wireless name =SSID.. same password and WPA2 AES (or Personal).
They will exist happily together but unfortunately if you use Yosemite, a connection to the Asus might get you ethernet but it can fail to find the TC.. but see how you go.. there are work arounds.