Using a backup file to transfer a site to new directory I am working on a project on a sub-domain of my site that when it is ready I want to move it to my client's site. I was wondering if I can do this by creating a back-up file using 'Backup and Migrate' and installing Drupal on their site and loading this back-up file to move all my work there. Is this possible or is there a better way?

All the best!
This will depend a little on which version of Backup and Migrate you are using.

Older versions only backed up the database, so you would also need to copy over the public and private files to the new site, along with the rest of the code.

Newer versions can backup files, but I don't have any direct experience with doing this.  It advertises that it can do either the public files, or the entire site.

Just keep in mind that Backup and Migrate will by default exclude the data from certain tables, mainly the cache files, logs, and the search index.  Rebuilding the search index can be a very expensive (and lengthly) process on a big site.  This is configurable, though.

There are other options, too, such as drush.