Best connection method?

However, when my iMac backs up to the TC, everything slows down and the TV pictures (through the BT TV boxes) go 'blocky'.

There is no Quality of Service in the Mac setup.. unfortunately backups are rated more important by your computer then streaming video which should always be the top of the QoS heap.

The issue is also that EOP does use common bandwidth.. unlike ethernet which is based around a switch which is able to direct packets to multiple devices at the same time.. neither wireless nor EOP can do that.

better way to connect the TC to the iMac that won't affect the network.

Ethernet.. now and always.. and forevermore.

I thought backing up wirelessly was quite slow, especially when a full backup as being done, but I don't think there is any alternative - without splashiing out on a Thunderbolt or other (Firewire?) type of drive.

How old is the iMac?? If it is 2012 or later it has USB3 ports which are both fast and cheap.

BUT.. even USB 2 on earlier computer is still faster than gigabit over network. Even if the USB is slower on paper.. it reality it works very well. And a self powered drive 2.5" drive can be plugged directly into the mac and pretty much hidden totally from view.

You can do occasional backups to the TC.. and keep using the local drive instead.

Otherwise you can wire the house with ethernet.. an expensive but total solution to the problem.

I didn't think I could

Correct.. you cannot connect a TC to the computer by USB.. it is purely for printers or hard drives which are read and written by the TC.

would I have to buy another drive - something I'm reluctant to do.

Not sure why you are reluctant.. if you want a central storage system.. ie a proper NAS.. the TC is the wrong product.

If you want to speed up networking without going to the cost of ethernet.. local drives are cheap .. and they are faster and far more reliable than backups to the TC. I recommend Toshiba or Touro or LaCie (rather than WD or Seagate) etc 2.5" USB drives.. They are fast enough, cheap enough and easily self powered on modern USB3 system.

I see too many reports of issues.. They seem to start out well but have gone downhill.

I like the idea of using the fastest technology (Thunderbolt) I can, but I'll see what other drives are available.

Bulk hard disks are much slower than USB 3 .. in fact even ssd drives are slower than USB3 although there is some latency.. so there is no advantage in thunderbolt unless you are using RAID SSD.

And the USB is many times cheaper .. thunderbolt cables alone are like $60. And the disk enclosures are very expensive before you even put a disk in it.

Thunderbolt is a great idea.. that needs an order of magnitude drop in price before it hits the mainstream.

Backups use slow spinning drives.. low power and low noise and thence low speed. There is no need for more.. still faster than Gigabit ethernet.