How can I "clean out" my Time Machine backups if Time Machine won't backup because the disk is full?

Solution 1:

Enter TimeMachine, then in the main window, select the backup you want to delete (on the right), then use the Action Menu (Gear icon) to select "Delete Backup".

Using this menu, you can also, when a file or a folder is selected, delete all its backup.

Solution 2:

The other answer didn't work for me, perhaps because I was trying to delete extra backups from an inactive TimeMachine folder. (I don't get a "Delete Backup" option in my Finder dropdown, as shown.

In Mountain Lion, I was able to use tmutil, a terminal command with great power. A great description of how to use it is http://blog.hawkimedia.com/2012/08/reclaiming-a-timemachine-volumes-disk-space/