BSOD doesn't automatically mean hardware problems. (I can't think of a reason a router would be causing that kind of problem, but God only knows.) I definitely wouldn't go buying a new computer until you try to format it and start over. Reinstall Windows and don't update any drivers at all and see if you still BSOD. Then start upgrading drivers one by one over time like Yamaneko said.

Personally I would skip trying to figure out the problem. You can backup everything important and reformat and reinstall Windows in a matter of hours and it's guaranteed to fix everything (for sufficiently low values of "fix"). You can spend the next month trying to figure out what the problem is and very likely never find an answer. Windows isn't meant to be fixable or even decipherable. Over a year without a Windows reinstall is pushing it.