You are going to have to fix the problem with the video first. It shouldn't be doing that. Reset BIOS to defaults and check Power Saving options (disable them for now). Be sure its not the monitor (try the other one). If anything, the monitor should stay on forever since the OS isn't running yet.
I'm pretty sure the video card or motherboard is damaged. You should really have a qualified tech check it. You should be able to run Dell Diags on machine, and it will run forever. Monitor will stay on and all checks should report OK.