Assuming you are using the correct size punch for the chain you're breaking, you should be able to get it to pop the rivet out if the punch is centered. However, if it just isn't working out for you, you might want to pre-grind a bit of the rivet head before punching, so there is less material for the punch to overcome.
I am assuming that you have the chain in an appropriate holder for punching the rivet out, and not setting it on some solid surface where the rivet cannot be expelled out the backside. This seems so common sense, but I've seen someone try to punch a rivet out against a solid surface before...figured it couldn't hurt to mention it.