How to perform a full system dump/backup with a broken screen?
Not a full-fledged answer, as even an excerpt would make it a rather long post, but here are some things you could try:


An XDA article describes How To Backup and Restore using tar on adb shell. The article holds complete scripts for both. Still you should not execute them straight ahead as they are, but first use e.g. the mount command (without parameters) to make sure the mount points are correct, as they might differ between devices.
Use AndroidScreencast1 to "transfer the display to your PC". This way you see what you are doing, and might be able to backup using apps like Titanium Backup or Helium Backup
GadgetsDNA has a useful article with 15 Useful Android Terminal (ADB Shell) Commands2, including how to restore a Nandroid backup via fastboot. Unfortunately, nothing to create such.
A German Tutorial at Android-Hilfe.DE (GoogleTranslate Variant) describes a tool named ADBRecovery, including download links. This should enable you to do a Nandroid backup as well as restore.


I hope one of these hints proves helpful to you.



1: original code seems no longer maintained; yankee found this maintained fork working fine
2: links to Archive.ORG as the original page is gone; you can also find it at Slideshare