Vista - Problems with both backup and disk check

I have an external hard disk which I have backed up to once already. I want
to start from scratch with a whole new backup. However, I'm getting
'catastrophic backup errors', and when I attempt a completely new backup I'm
told that the C: drive has errors. I then run disk check following the
instructions given in the help menus (and online). C: is my boot drive,
though, so I get the message that I can't run disk check while the disk is in
use and must schedule a disk check on restart. I click on this option and
restart (manually) but the disk check does not start.

So - that means I have two problems: firstly - disk backup is failing
because I have errors on C:. Secondly, disk check won't start. So - I'm stuck
with a computer that won't backup. (sigh) I miss the good old days with the
XP version of backup where I could specify full or incremental backups, and
everything worked simply and well.

The stats: Dell Inspirion E1405 (Model MXC061) running Vista Home Premium,
Intel(R) Core(TM)2 CPU T5500 @ 1.66 GHz, 1014 MB RAM.