Enable screen sharing on mac w/o display
Your going to have to get into your machine to set up screen sharing. The only way that I can figure is to take the drive out and place it in some other machine.  I suggest placing the drive in an external drive enclosure.  I set my mac mini machine up with a static ip address. Once you get this to work, you would move the drive to the old machine.  

I use Realvnc client and Realvnc server.  They have a free version, but may be slower than you want. https://www.realvnc.com/en/connect/docs/faq/pricing.html I pay for the local lan version.  I turn off encryption.  Not seeing a connected display, vnc will use software drivers.  You'll need to reboot about once a day as the display will get messed up.  For those with working video hardware, you can attach a dummy hdmi plug. This will speed things up a bit and avoid display mess ups. I'm connected to my mini via 100meg ethernet.  Works well enough.



Does this machine work in safe mode?  

You may have a problem with the video chip.

Here is how to tell...
  Shutdown your machine.  Hold down the shift key.  Poweron.  Bootup will be longer.
    Wait awhile while you harddrive is being checked.
  An article on how to boot into safe mode.
    http://support.apple.com/kb/ht1455

Runs OK in safe mode...
Running in Safe mode leaves out some video drivers.  Which results in your machine not using advanced video hardware. As luck would have it, you can run the safe mode video drivers in normal mode. 

Here is how:
https://discussions.apple.com/message/16057567#16057567

Look through the above thread.  See the second page.  You don't have to read through the first page.  Just go to the part where I try a solution that works.

Using the safe mode video driver results in these limitiations:
  -- OpenGL acceleration is not supported on this mac
  -- Your mac lacks quartz extreme acceleration

Summary of G5 problems. Includes instructions on a hardware fix for the brave at heart:
See -Rotten Apple- articles in both of these threads.
https://discussions.apple.com/thread/4023152?tstart=0
https://discussions.apple.com/message/18700825#18700825

Robert 

R