A virus would not cause something like this. This is an issue that is occurring before a virus could be able to execute, or anything else for that matter. It could be the screen or the screen cable or something of that nature I suppose, but being as it froze up before this happened, yet the screen was still on, then that's not likely either unless only by coincidence. As far as the BIOS key, those vary so widely that you would need to look up your model to figure out which one exactly it is. It can literally be any of the F- keys or Delete or even Escape in some cases. For HP, you can usually hit Escape and it will list what F- keys do which function. It does sound like a deeper hardware issue though.
The next thing I would check would be the CPU, and by check (in my case) I mean grab a compatible replacement out of the drawer and see if it boots with it installed. This usually requires a full teardown of the laptop unless its one of the few nice models that allow removal from the bottom, or otherwise by removing the keyboard and taking it out the top. This is probably the point at which you need to decide for yourself how much of a technician you want to become in a short amount of time lol (Because based off all the info here, we've prettymuch narrowed it down to either the board or the CPU). If its just a bad CPU, its usually no big deal unless its a BGA type, which means that it is one with the motherboard and inseparable without using a heat gun and really knowing what you're doing. I would not recommend this to you because it can be difficult and its usually not worth it. On the other hand, if its a traditional CPU with a ZIF socket (removable), then buying a cheap replacement might be a good option. You'll need a bit of thermal paste BUT NOT TOO MUCH (common mistake among CPU newbies). Just a bit more than a drop of the stuff will do the job for a laptop.
If you decide to try this and it still doesn't work, guess what?!? It was the motherboard! (those can be replaced too, but again, how much taking apart are you comfortable with doing?)
Oh hey look, another wall of my text! Sorry.....
tl;dr Sounds like a bad CPU or board.