Try installing just one stick of RAM in the 1st DIMM. (Depends on the motherboard which slot is to be used). If it still doesn't go to the BIOS, try a different stick. Have any spare compatible RAM? The idea here is to see if a bad stick of RAM is causing the problem, trial and error.
Also, even before you do that, trying clearing the CMOS. Take the CR2032 battery
off of the motherboard, wait a few minutes... and press the power button a few times, holding it down sometimes to rid the motherboard of any electricity. Do this with the computer unplugged obviously. Replace the battery and then try getting to the BIOS.
I would recommend that you unplug EVERYTHING that doesn't need to be plugged in, i.e. hard drive, cd drive, case fan(not the cpu fan), floppy drive, etc.
If any progress is made, start adding one component at a time powering on after each addition. This will find what is causing the problem. Trial and error.
If none of the above works, I would suspect a bad motherboard, or cpu... in which case it's gonna be the motherboard 90% of the time, cpu's really don't go bad that often... and usually outlast the motherboard.
Let me know how things go.