This worked for me: Mix up some mild dishsoap and warm water. Lay your bike down so gravity will help you. Take an eyedropper filled w/ the soap & water and slip in between your grip and handlebar. Drip a couple drips at a time in there as far as you can get. Squeeze and twist your grip. Repeat until grip loosens. All in all I only used a few drops ... The bigger trick is getting the grip to dry out again and not be slippery once moved.
When loose I took the grip off ... dried the bar and the grip. Then used about one drop to slide on where I wanted it. That way it wasn't too slippery to dry out soon. Some people recommend hairspray for the reapplication of the grip. I think it is moist enough to slip on, dries quickly and with just a little stick to it.
Good luck!