As of OS 10.7, you can control Time Machine from the command line. The command line tool is called (and is called with)
tmutil. Open up the Terminal, which is located in your
/Applications/Utilities folder. It will be easy to get there if, From the Finder, you type Shift+Command+G, which brings up a sheet where you can type the destination path to the Terminal (copy and paste this):
After the Terminal opens, and at the prompt, type
man tmutil, which will get you the manual page for
tmutil outlining all the commands and options available. It's quite extensive, and more than can be covered in a single answer, but examples might be:
Turn on Time Machine backups:
Set the destination to be used as backup:
tmutil setdestination... (this has options that can be added to specify destination information and type)
Start a full backup to a drive:
tmutil startbackup -d /Volumes/Name_Of_Your_Backup_Disk
You must carefully read this manual all the way through and understand the terminology before beginning. The manual is also available online at Apple's manpages reference area. You can read it here, and there is a good primer on Simon Heimlicher's site with much better examples than what I've given above.
As with all command line tools, USE CAUTION. You will have to run most of these as root as well, which means that you're granting yourself power to do lots of potentially nasty things to your computer. In the case of
tmutil, though you're unlikely to do much damage as you have to correctly specify a destination before it'll actually do anything. Probably the greatest hazard would be if you'd already set a destination you could accidentally erase it; since you have no destination, though, that's not going to happen.
Good luck getting your backup going and with getting your machine fixed!