I have a Toshiba satellite c660/c660D. This is what happens, I plug it in, turn it on then after a minute the fans go crazy and the laptop turns off. I only try to go into the bios as i have no hard drive in it.
The laptop maybe overheating since it's also happening while inside BIOS as you mentioned. Do try to open your laptop and remove the dust off the fans and apply new thermal compound on the CPU specially.