If it chimes but you don't get any image even on external monitor there are two possible issues, either it's unable to boot or the graphic card doesn't work. Either of the two is a serious issue involving one of the main chips and needs a deeper level diagnostic. Your machine was one of those subject to Apple Repair Extension Program for Video Issues, but unfortunately that's expired. You'll have to look for a top level repair lab or a replacement logic board, I don't see any other alternative.