First thing to do is isolate the projector from the rest of the system. You want to find out if the projector is the cause or if the problem is in the distribution system or source.
Use a different source device (DVD, PC etc.) and use a good short cable and go straight into the projector. Disconnect everything else from the projector except for the power cable.
If the image comes up, this means the projector input is working and you should then add the next part of the distribution system.
Keep adding parts of the system until the problem comes back and that is where the problem is.