BIOS Problem - Stuck at Splash Screen

Computer is stuck booting at splash screen. It's an HP desktop. Cleared CMOS and real time clock and I was able to get into BIOS to change some settings but when I rebooted, it got stuck again.

Advice?

I would try booting with one stick of RAM and with minimal USB devices.

Replace the PSU if possible (For testing)
As Tyerker said - 1 stick of RAM, nothing plugged in.
Check the motherboard for blown/bulging capacitors.

This computer is so frustrating. I'm having to clear CMOS every time if I want to get into BIOS. It will get stuck at the splash screen if I pause it at all. I'm working on this thing for family and the original problem was that it wouldn't boot to Windows. It's starting to look like the hard drive has failed. It shows up in BIOS but I can't mount it in Ubuntu to try and recover data. Unity can't see it and TestDisk can't see it.

try removing everything from your computer, leave 1 stick of memory, keyboard and monitor plugged in. does it boot? add hard drive, does it boot? step by step logically through it and see what happens.

I think you misunderstand. The computer boots, but it gets stuck on the splash screen. At any rate, I was able to change the boot device and get into live linux. Unfortunately, the hard drive is definitely dead.