Screen turns on then goes black but iMac is still on.

The most likely cause is the backlight board. It'll go on for a couple seconds, then essentially get tired and turn off. If it truly is the backlight board, if you look really close right after the screen shuts off (like really super close, and possibly with a flashlight), you'll see a faint outline of the Apple symbol in the middle of the screen. It's an easy enough repair if you have a replacement and the right tools.

Other sources of your problem include the actual LCD, and the graphics card. It's harder to test those though unless you have another one you can put in to see if it fixes the issue.