You can usually check that all is ok up to the power connector by borrowing the connection to the monitor. The two leads are generally similar. The monitor generally has a power on indicator light which changes colour when the monitor is active, so you can use this to confirm that there is power getting through.
You can easily bypass the computer power switch if you are bold enough to touch the power switch pins on the motherboard. If you can identify these pins, simply short them momentarily together.
The only way to really check the power supply is by substition. A cheap PSU is not much more than £5, and is of surprisingly good quality for so little money. You need to make sure that the power rating is sufficient, but this is a 300W supply click here, and this one is 500W click here.