In your top post you said the computer was shutting off. Is the system working properly now or are there still problems?
If there are still problems it would be best, at this stage, simply to copy over to some external drive your personal files and data so that they are preserved should the OS go south, (fail). If Vista works then copying is just a matter of connecting the external, opening two instances of Windows Explorer, with one window opened to your external and the other where you navigate to where your personal files and data are and then 'dragging' those files across to the first Windows Explorer window. In this case, (internal drive to external drive), dragging leaves the original file where it is and creates a copy onto the external drive.
As to a full backup, firstly may I say, never use Windows' own backup feature. It rarely works the way we want and these forums are full of instances where it has failed.
However, your Seagate drive should already have backup software installed on it. If you look at http://www.seagate.com/external-hard-drives/portable-hard-drives/standard/goflex/ and the basic description under the "Compact and powerfully flexible", you will see where it says "Preloaded with easy to use backup software".
But there is more. Because this is Seagate, you have access to their "Disk Wizard" software which includes a version of Acronis TrueImage. I assume that is a licensed version to Seagate. I have downloaded that myself and use it to create complete system images of the OS drive. These 'image files' can be stored on the external drive as many times as you like. The Acronis software also allows you to create a Boot Disk, (CD or DVD), which you can then use to re-load a hard drive with one of the image files.
For that, just go to Support in the Seagate web site, then Downloads, then Disk Wizard. It is all free.
So, no need for Norton Ghost or any other software if that was what you are looking for.
Having used Acronis a few times now I am getting used to it and its Help files are very useful, if a little confusing at first. But recently when my Win 7 failed, I used the CD I created to install my latest backup onto the drive with no problem and it took about half an hour or so.
You mentioned in your top post that something wouldn't back up from the C drive. What more can you tell us about that?
Mark