|The fact that the monitor turns on has little to do with the state of the power supply even if the monitor does take its power from a socket on the computer. That is just a split-feed from the mains input and has nothing to do with the power supply.|
This is assuming that it is a desktop and not a laptop. Even with a laptop it could still be the power supply at fault.
Without any system specs any idea of what could be the problem is will just be speculation.
Do you get anything on the monitor at all? Any bleeps or lack of them. If the monitor is completely blank and the computer is completely dead it is almost certainly the power supply.