The screen does not stay on. It keeps on turning black...

You will have to look inside, first place to look will be the caps in the power supply section, look for leaking/bulging tops/seals.
If you are going to DIY and already try the monitor with another PC, and have proper tools and know safety precaution then please read on:
Most common failures in the LCD monitors are bad capacitors (bulging top/seal or leaking) in the power supply (they should be replaced in a set), blown fuses; poor solder joints, failed inverter circuits (blown fuse, shorted transistors, shorted/open transformers), bad lamps (poor solder connections or worn out lamps). You will need to open it up and inspect the inside, see example of failed VIEWSONIC monitors to get some ideas what to look for: http://s807.photobucket.com/albums/yy352/budm/
Post back what you see inside so we can guide you further and it will help out other people in the future also.

Basic LCD monitor and TV troubleshooting guide:
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide
http://www.fixya.com/support/r7406380-tv_training_manuals
Failed TV and Monitors: http://s807.photobucket.com/albums/yy352/budm/

Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague

Capacitors kit: http://lcdalternatives.auctivacommerce.com/ he can make you a set of caps for you.
Or www.digikey.com just make sure to use caps with low ESR, 105c, high ripple current, long life rating such as PANASONIC FM or FC series.