I've tried using a polish cloth, but the chain is too fine. I don't want to damage the delicate chain - what should I try?
Real gold plate won't tarnish. If it has a high proportion of silver that has tarnished, that tarnish can be removed electrolytically with no harm to the chain:
Make a solution of about ten gram of baking soda in about 100 ml water, place in a disposable aluminum foil pan, drop in the chain so it rests on the foil, and warm gently (no need to boil). Tarnish will disappear in a couple of minutes.
Of course, if it's cheap metal covered with yellow lacquer to look like gold, this won't help, and may remove the lacquer.