Should I grade my property or put dirt against chain link fence?
I would shave down the high side (left side in picture) from a distance of 2 or ft and use that to fill under the fence. Assuming the fence is on the property line this would transfer some soil to your neighbor's property. You could assume he won't mind, but it would be best to discuss it with him first.

If this fence is designed to contain a dog who is or might be a digger, you might have to place cobbles or concrete "stone" blocks with decorative side toward the neighbor, under the fence and fill on the back side. Of course, it might look strange to have blocks only under some sections of the chain link fence.