On Friday my monitor broke, and I couldn't get a picture from my PC or my Xbox. I sent it off and it was replaced. I could still get a picture from my Xbox, but using the DVI cables I had the monitor couldn't pick up a signal. I rang up my CS and they told me to use a VGA adaptor and it works, but there is a major green tint to my picture and the resolution is a bit dodgy. I ordered some new DVI cables but they can't pick up a signal either. What could be the problem?

And your monitor is set to accept digital signals? For instance, there are several OSDs that allow you to plug in multiple cables at once and you can switch between them using the settings. If so, then a broken DVI input could be one explanation.

I assume so, the replacement monitor is the exact same brand and make and I haven't done anything different to set it up than with the original one. If the DVI port is broken, do I need to ring up the company that made my PC and get it fixed?

Provided that you bought it from them, yes.