Then we have established the issue is NOT specific to the graphics card, BUT IS specific to the parts remaining (PSU, CPU, RAM and Mobo).
So as we don't know what parts are working or not, lets make sure
A) use another HDMI cable from onboard to Monitor
B) use another monitor / TV to see if the Monitor is faulty
Then comes the fun part... Get your Motherboard MANUAL and look up BEEP CODES.
Remove both RAM chips, press power, do you get the Beep code for problem with RAM? If NO then problem with Mobo or PSU
If Yes, put back just ONE RAM, Remove CPU, press power do you get the Beep Code for problem with CPU? If no then Mobo or PSU
At that point you will know how much of a problem you have (mobo or PSU) and would need to replace out at that point.
Now you can try a new PSU, but when you get down to these components you will need to consider maybe replacing out more, especially if the rest of the PC (RAM, CPU, MOBO, etc.) are OLD hardware that is will be end of life anyway (more then 3 years old).