OS X Startup Disk full, cannot login
Deleting files in this circumstance is relatively easy but you will need to be at least a little conversant with Unix/Linux command line operations.

Boot your Mac holding down Command ⌘-S, this will pop you into single-user mode. A black screen with white text scrolling by. Eventually you will get a prompt with some instructions on what to do directly above it. I don't remember exactly but the commands mount the filesystem so you can access it. Then enter this:

cd /Users/[USERNAME]/Movies


Where [USERNAME] is your short username that appears on your home folder. Once in the movies folder you can proceed to delete movies. remember a couple of things.


The command line is Case Sensitive. So MySpecialMovie.mov is a different file than myspecialmovie.mov. At least at the command line level. (The Mac preserves case but in Finder operations is not case sensitive)
The Tab key is your friend. Press Tab when typing a file name and it will autocomplete.


Start by listing the files in your ~/Movies folder like this:

ls


Or you could get a longer listing (which I prefer).

ls -al


That will give you a list of all the deletable files. 

The second method will give you file size. 

To remove one of the files:

rm MySpecialMovie.mov


Now it is a matter of going through the ~/Movies folder using ls and rm to view and remove un-needed files.

Note that pressing the Up Arrow will scroll back in command history allowing you to easily review what you have done and/or repeat previous commands. Down Arrow scrolls forward in command history.

One you are done you can issue the reboot (restart?) command and allow your Mac to boot normally.