..just a mineral spirits soak and shake up in a little container or a bottle until it runs clear and a final alcohol rinse to get any residue off will do. If you are stripping a well used chain you may just need a couple more rounds of the spirits to get all the crap out.
Where to dispose used solvents should be listed online somewhere for where you live.
If you are buying a new chain anyway, just source a prepped and pre-waxed chain to get you started. If you don't already have the solvents and want to save yourself the disposal trip, it's usually well worth the service charge. You won't have to solvent clean it again thereafter.
If you have a waxed chain that is due for its next immersion, it's enough to just wipe it with a microfibre cloth to remove any surface dust, or if you have it on hand, just a few sprays of isopropyl on the cloth, run the chain through the cloth a few revolutions while on the bike, then take the chain off and drop it into the wax. If you set up a little raised sieve in your wax pot that keeps the chain from the bottom where flushed out contamination accumulates, you can even skip any chain wiping and just chuck the used chain into the pot.
So I just need some mineral spirits (and I can look up where to take this for disposal), and maybe some alcohol. Or, I can start with a pre-waxed chain for a little extra cost. Oh, and find or make some sort of sieve for it to sit in while soaking in the dedicated crock pot or saucepan, which have to be stored when not in use. And make some arrangement for the chain to hang from while drying/cooling (preferably above the crock pot/pan). And when it's time to re-wax, all I have to do is remove and reinstall it on the bike. Not a problem, since quick links are not very expensive, and some of them are even re-usable...
You're right. This is much faster and easier than wiping down the chain with a rag once a month and dripping a little lube onto it. I don't know why I didn't pick up on this process long ago.