Before You Begin
Give your camera warranty a full reading. Has it expired? Does your camera qualify for repairs? And of course: Will it cost you any money? Though you'll probably have to pay shipping, having your camera shipped off to the factory for repairs and replacements is probably the cheapest and most guaranteed fix job you can get.
Even if you can't get it shipped to the factory for cheap-cheap, calling the technical support line is another good idea. We all hate the long waits and often repetitive advice, but they often have useful advice that is specific to your camera that might not involve an invasive repair job.
Reading through the troubleshoot section of your camera manual might also be of help. If you've lost or otherwise misplaced your camera manual, it is often the case that they are available online, or that the company website will have an even more detailed troubleshoot manual on their website.
If none of these provides a cheap, easy fix for your camera and you know your way around a screwdriver, you might want to consider doing a DIY job. Please be aware, however, that it is likely to ward your warranty if it hasn't already expired.
If none of the above at least gave you an idea of what's wrong, you'll need to determine what precisely the issue is. What caused the LCD screen to stop working? Was it the result of gradual damage over the course of its lifetime, or was it a sudden, discrete accident? If there is obvious damage to the screen itself, say, a large gouge or an edge peeling off, then chances are it's an external problem and the screen itself will need to be replaced. If there is no obvious damage to the exterior, then chances are that it's a problem with the circuitry inside.