It seems to me that if you paint a chain you'll kill the main benefit of a chain with links... flexibility. The paint would probably seep into all the links and you'd end up with one very stiff (and unusable) chain. Now, if you had incredible patience, and lots of time on your hand, you could probably hand paint it, one link at a time taking care to not get paint between the pins and plates. Starting with a new chain, completely degreased you might try automotive paint.
Any reason why you want to paint it?