In an engine it needs 3 things to run, air, fuel and spark so as it is cranking it has to be an issue relating to one of the 3 things so you should check if you have them 3 things and go from there
When the monitor shows no display does it turn off the whole pc and need turning back on? If the whole pc shuts down it could be a faulty PSU or an over heating cpu/gpu If the whole pc does not shut down it might be a faulty gpu, it could also be an outdated gpu driver and could need updating
My guess from what u have said it sounds more like a complex issue and depending on the age and make of the tv it might be cheaper to buy a new one over trying to repair it