FYI, things aren't always as they seem, and it could still be your PSU (just because it powers up, it doesn't mean it's in fully working condition), if you have a spare PSU (or another computer to take one from for testing) then I'd suggest you give it a go. But apart from that, other possible reasons include improper BIOS settings, bad RAM, or a bad video card. Try the following:
1. Reset your BIOS settings - Unplug your computer's power cord, open up the case, locate the round (CR2032) battery on the motherboard, use a small flat screwdriver to carefully remove it, then put the power cord back in, power up your computer, if you see text on the screen, then it means it worked. Then power it off again, unplug the power cord, put the battery back in the motherboard (in the same position as it was before, typically this would be with the plus sign facing the outwards), plug the power cord back in again, power it up, and modify the date/time settings from the BIOS (as well as the boot device priority if needed) as all of those would be resetted.
2. If you have multiple memory (RAM) sticks, try each of em standalone (with the other's disconnected) and power up the computer just with that one stick. It's possible one of em has gone bad.
3. If you have a dedicated video card, try physically removing it and using the integrated graphics instead.