But assuming that this computer is a Desktop and not a Lap Top you can start off by connecting another monitor and seeing if that makes any difference of if you don't have another monitor connect this monitor to another computer to test it.
Now assuming that the monitor works you should then replace the Power Supply with a Known Good one and see what if any difference there is now.
Many times a Power Supply in the process of failing will run fans show lights on the front of the case but it will either not be providing power on the 1.3 V Rails or the voltages will be correct but the current will be to low to actually power on the computer. This is quite common if you return to a computer that was left plugged in and turned off when there was a Blackout or other type of interruption to the Mains Power.
If replacing the PS doesn't have any effect you can try another Video Card if you have Plug In Video Card and if you have On Board Video you need to replace the M'Board.
Col