Backing up to External HD over WiFi

When you use any Network Disk as a backup destination, Time Machine will create a sparse bundle disk image (a Volume inside a File) to store the backup. So you can not use your existing backup drive "as is".

The late James Pond wrote a wonderful set of illustrated, approachable articles on Time machine related subjects, and had a lot to say about this setup. He said it is not reliable, and if you use it, you should have another backup method as well.

.Pondini's Time Machine - Frequently Asked Questions -- 2. What can Time Machine back up, and where can it put it’s backups

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Grant is quite correct - it is possible to make this work for a time but it isn't just unreliable - it will break. There will come a day when you get the error message that TimeMachine was unable to complete the backup and you'll never get it working again short of erasing the backup drive and starting over. I don't know what kind of magic Apple used in its TimeCapsules that made them reliable but whatever it was, I've not yet come across another router that has it. You may find that another backup program will work reliably with a NAS but I eventually ran an Ethernet cable from my wiring closet to my desktop to back up to my NAS - not with TimeMachine though. I used another backup program.

Most folks are using a cheap NAS, which is a small built-in computer inside a drive enclosure (usually running linux) that makes the drive appear as a station on their network.

You can hang that on Wi-Fi if you insist, but Ethernet completes the transfers faster and does not slow down your web-surfing much. Of course you could go Ethernet from one of your Routers and Wi-Fi computer-to-Router (or any other combination of wired and wireless) if you prefer.