Well if you have a backup then restoring it would be the simplest solution. You can actually see the backup from CWM recovery? You don't say why it fails.
But OK, in that case let's try flashing the ROM again. Firstly, what version of ClockworkMod recovery do you have? It will say at the top of the CWM screen. Just need to know that it's sufficiently up to date.
Secondly, assuming you wish to use the CM ROM (because you've already downloaded that), check the file is intact. You do this by checking the md5 checksum of the file matches the one given by the site you downloaded it from. On Windows you'll need to download some utility to check the code (I don't use Windows, so can't recommend). With Mac you just open a terminal and type "md5 filename", and on Linux it's "md5sum filename". If you have doubts, or don't have the md5 value (e.g. because you used something like ROM Manager to download the ROM) you can download the current stable CyanogenMod for the International HTC One (m7ul) from here:
CyanogenMod Downloads - just pick one labelled "stable" to be ultra-safe.
Once you have a ROM on your computer, connect the phone to the computer via USB and check that ADB can talk to it by using the command "adb devices". That should return the serial number of the phone. ADB should be able to talk to the phone while it's running a custom recovery.
If that works, you can then copy the ROM to the phone using "adb push":
adb push yourROM.zip /sdcard/.
Once it's copied over, you can then do a factory reset (just /data and /cache, not /sdcard or /system) then "install zip from sdcard", select the zip you just copied over, and install.
With CM you'll also need a second flash to get the google apps, but let's get this bit done first. I assume you also downloaded a google apps package, but it needs to be the correct one for the android version, and as I don't know which CM version you tried to flash I don't know which that should be. So getting the ROM going is the first priority.
If the above fails we can try "adb sideload", but I've never done that myself so am suggesting this way first.