The easiest way is to use a Dremel drill with a very fine metal drill.
Take your time, the hardest part is to keep the drill straight.. Once all the metal of the broken screw is out, you can now insert a new screw.
Should you F... it up, use a larger drill, no oil, drill it all out and refill the hole afterwards with "liquid metal". Let it harden out, and drill a new hole in the hardened filler for the new screw.
TIP: to avoid broken screws... screw it gently in... once you feel resistance, screw it a bit out, and screw it again and a bit further, then go back again etc till it's all in. You should never screw it all the way in in one time. Forth and back with some oil is the safe way to go.