How can I tell Time Machine to keep using its former backup? For some reason Time Machine doesn't think that the backups on my external drive are for my current computer.

How can I point Time Machine at the former backup files so it can pick up where it left off?
On OS X 10.7 Lion or later, yes. You need to look at the tmutil command, specifically the inheritbackup verb (which tells Time Machine that this is a backup of this machine, and not some other), and the associatedisk verb (which tells Time Machine that the disk it was backing up before is this other disk that you have now).