After the BIOS update, how far does it go? Do you only see a blue screen soon as you turn on the PC or after Windows tries to load? Can you enter the BIOS after starting the laptop? Does Safe Mode work?
If you can't see anything on the laptops screen after you turn it on, your BIOS is corrupted and you will need to replace the BIOS chip, which is not very cheap, but cheaper than a new laptop.