Actually for me, number 1 looked easier. Depends how much you want the data - there's a reason why "data recovery" is one of the highest-paying advertising phrases on Google.That's it, the thingy-mabob.
Saw on number 1 that you might be able to do it without one but it looked complicated. I'm crap at tech stuff. The second version looked almost idiot-proof, which is where I come in.
If your old computer is not working you can't get to the user interface where you tell all the pixels and twiddly bits what to do. So you need to take the bit which hopefully still has your data (the hard drive) and put it somewhere where a computer with a user interface can read it. Installing it on a new computer (not laptop) sounds easiest.
If that don't work, unscrew the hard drive, take it to a computer shop and ask around until you can find someone who can retrieve the data and stick it on a stick or an external drive for you.
(You know that you will be backing up in future, but for the sake of anyone who is reading this and isn't backing up their data regularly, do it or feel a **** when you lose it.)