Op-amp astable multivibrator with single supply
What would I have to change when I want it to work with dual-supply
  (currently it would kill my led)?


Put a reverse protection diode across the LED (or maybe a reversed LED of a different colour to get a nice flashing two-colour effect).


  How do I need to change the circuit to get it work with single supply?


Try this: -



The frequency might need a bit of an adjustment if it's critical.