Haven't tried the burnishing fluid on chains yet... that could be one approach; that would 'blacken' it, to which you then apply weathering, or areas of vehicle colour, the shiny bits having been removed with the blackening (sort of a reverse weathering)
I don't have an airbrush; my current method is 'messy': to pull it through a shallow container of thinned paint, or load a broad brush with paint, and pull the chain through the bristles lightly wiggling, pressed a on a flat surface; might have do to it a few times to get all the surfaces painted.
The trick is to not have the paint so thick that it fills in the links.
Starting directly with AK 'track' weathering works for some situations.
I let the chain hang to dry. If it is going to hang in the model like a truck gate chain, I fold it before it dries as close to the shape as I can to avoid cracking the dried paint.
And for chains laying flat on a surface, you could glue it in place and paint it afterwards if it is the same base colour as the vehicle. A bit of wash, weathering, or highlights will accentuate the links.
For heavy weathering, I add pigment powder to the paint and daub with a hack brush.
Super tiny chains I glue, then paint in place with a fine brush.
A 'loose', heavy chain painted and weathered before placement:
