Is this a brick chimney with a fired clay liner? If so, the entire mass should just be masonary. The only thing that might contain asbestos is the furnace cements that seals the flue pipe to the chimney base.
Should be no great issues, just lots of dust and hard work. Start removing bricks at the top (of course that should be obvious, but I take nothing for granted after some of the things I have seen over the years), and work down. The floor joists, ceiling joists, etc will be headed off around the masonry work, so you will just need to fill in with some short pieces when finished.
Invest in a decent respirator, more efficient and comfortable than those aggravating dust masks, and a couple hundred bucks for a small Bosch or Hilti demo hammer (recommend the Hilti personally) with a couple of chisel blades will make easy work of breaking the mortar joints.