Normally if there's a problem at startup, the POST will beep its codes to tell you what's up. If it doesn't do that, the problem is very early in the boot sequence. Certainly, the BIOS isn't likely to be the problem (and you couldn't get to the BIOS to flash it anyway).
You say your monitor doesn't come on. That implies the graphics card isn't starting, and that's very early in the boot-up process.
Is the fan in the power supply working? They are early candidates for clogging with dust, and seizing up. The switch-mode stuff then overheats and pops its clogs. No PSU fan = probable PSU fault.
I've fixed about half a dozen computers in the past 12 months (I do this for friends) where the power supply had died. I even keep a spare power supply in the cupboard (which meant I was back on line within a few minutes last time mine died).
EDIT: do you happen to have a spare graphics card you can plug in, if the PSU fan is running? That's the next check I'd make.