How can I install a ceiling fan with an extension kit that doesn't have a ground wire?
Code doesn't require grounding for the fan because you're not really installing permanently in the structure, you're using it as a removable appliance.

If you want to ground it anyway, buy high quality aesthetically pleasing braided copper grounding wire, and weave it through the chain along with the original cable.



Connect it normally at the fan box, and at the outlet, either cut the plug off and add a replacement plug, or terminate the ground wire with a spade connector and use the outlet's cover plate screw to secure it in place.

You could improve the situation by replacing the outlet with GFCI or using an inline GFCI protector.  Both overkill if you ask me.