Blue screen on iMac

If your issue really is due to a full hard drive that prevents your iMac from booting then one convenient way to remove files is to boot the iMac into Target Disk Mode so that it behaves like an external har drive, then connect it to another Mac so that it mounts as an external hard drive that you can access and manage the files.

Use target disk mode to move files to another computer - Apple Support

Before doing so, you may want to try disconnecting all of the attached peripherals other than the keyboard and mouse to see if one of them is causing the iMac to hang.

You will need another Mac and connect the two together using Target Disk Mode and then move large libraries such as your music, photos and movies to an external HD. If you are not familar with TDM please refer to http://osxdaily.com/2010/04/07/how-to-boot-a-mac-in-target-disk-mode/

If you do not have another Mac available then contact your local AASP for support.

Finally, going forward NEVER let your Mac get within 15% of filling the internal drive!!!!!!!!!!!!!!!!!! For example, if your drive is 1TB capacity, only let it get to 850MB before offloading material to external HD's or deleting unwarranted data.