Assuming you can get to both sides of the door, hear is a useful video if you have a broken key in a rim cylinder lock.Most of the general public call it a Yale lock.
So to recap if you have broken your key in this type of lock and you can get to both sides of the door this is what you do.
1 Take the latch case off the door, this is held on with 2 or 3 screws depending which type you have.
2 Once this is off you will see 2 bolts these secure the cylinder lock.
3 When you have unscrewed these, the cylinder lock with the broken key in it will come away from the door
4 Slacken the circlip at the back of the lock.
5 The tail will now come loose.
6 All you have to do now is push a similar key or thin piece of wire into the cylinder lock key way and the offending broken bit of key will come out.Then just reverse everything you have done to have a fully working lock.
If you are in the West Yorkshire area and you need a experienced locksmith, you can call Absolute locks,we cover all of Bradford,Keighley,Shipley,Bingley,Halifax,Huddersfield.