Loosen the 3 screws on the bottom and pop the back off. Take a look at the capacitors. Many of the iMac G5's have problems with capacitors bulging/leaking and fail after doing so. They will look like small cylinders on the board. The bad batch had an "X" marking engraved on the top, the newer replacements (the good ones) have a "K" on the top instead.
http://en.wikipedia.org/wiki/Image:Badcaps-tayeh-4.jpgThe tops of the capacitors should be flat and shiny, not bulging up or cracked/corroded in any way.
When my video RAM went bad, I also saw similar problems, but luckily it was repaired under AppleCare. Try booting from the diagnostic CD (hold either C or D down when booting, forget which, different models vary) and see if you can make out the screen enough to run a test. My bad video RAM showed up as that (from what I could read) on the diagnostics.
If you have the bulging capacitors with the "X" I'd call up AppleCare and raise hell. These were a well-known problem and tell them that you feel that yours should have been recalled as well. Personally, my iMac G5 went through:
3 logic boards (1 bad video RAM, 1 bulging capacitors, and 1 wouldn't power on)
2 optical drives
1 front bezel (the place where the screws attach it to the back broke)
1 power supply (would not wake from sleep, then would not power back on for several hours)
A combination of A LOT of heat from the G5 in the small encosure and cheap logic board electronics made this model very problematic for a lot of people.