The obvious question is why are you replacing the PSU? Did the computer stop working or was it not performing well enough and you went for an upgrade?
If the former is the case it may be a dead motherboard.
If the latter is the case go back through the following (don't get annoyed if you've already tried it- sometimes overlooking the obvious wastes countless hours
because you're sure you've done it - but actually haven't
Make sure the wall socket is working as well as any extension cables and multiplug packs - test with a desklamp or similar.
Make sure the power cable is firmly connected to PSU
Ensure the PSU switch is turned on.
Ensure all connectors are firmly and correctly! connected.
You could also try resetting the CMOS by removing the battery for 10 minutes and replacing it (or shorting the CMOS jumper if available).
Is the new PSU "new"? Can you test it in another computer to ensure it works?
Failing all these the motherboard increasingly looks like the culprit