How to connect chain links to the axle
You probably want something a little similar to this:



There was a similar setup in the Bespin Cloud City scene that used a string attached to two studs that was wound around an axle in the same way.

As Phil B. suggests, adding an additional round plate on top of the chain will provide you with a bit more support for the stud, at the possible expense of making the winding thicker - it will tend to wrap around the plates then.