Generally, all PC monitors meet the same specifications needed to support basic PC video/graphics (VGA, SVGA, etc.). In other words, your good monitor should work with your good PC for basic video. If it doesn't, it means your PC, monitor or related hardware/software is bad or misconfigured.
I'd recommend checking the components of your unknown PC in your known good PC. Take out the drives and put them in your good PC. Check any other components that are user changeable such as PCI cards, graphics cards (especially this), RAM and cables. Do this one at a time and keep in mind that some stuff may not be compatible with your good PC.
Once you eliminate the good stuff as your problem, what's left is likely the cause.
Dell
BTW - many gaming machines die from overheated electronics, especially overclocked processors and graphics cards. There's probably a good reason why it was at the dump!