See my Apple Support Community User Tip:
The solution for "You need to free up space on the startup disk"
You will need at least three seperate external drives.
1: For a TimeMachine backup of your entire present boot drive, perform this first, will take some time.
2: A storage drive for moving some files off (Movies is a good choice of large files to free up space off the boot drive, just drag and drop copy, then Trash the originals)
3: Another storage drive for a backup of the first storage drive, as you need to retain two copies of your personal data at all times.
When you update the TimeMachine drive after deleting files off the boot drive, it will eventually delete the files you deleted off the boot drive, as it's a rotational in time backup system.
Use Activity Monitor > Disk Usage to keep your boot drive below 80% filled. You will need to reboot the computer for it to recognize the new drive space after your done copying and Trashing files.
This free program can help visualize where your drive space is going.