You don’t say what material the chain is made with!
That would make a difference to how it was cleaned. If it is silver
or gold, perhaps gently playing the torch over it to burn off the
muck, followed by a stint in a tumbler to repolish will do the job. I
cleaned a heavy twisted link silver chain that had been constantly
worn and never cleaned for years this way, and the chain looked like
new when it was finished. The burning off process was pretty yukky -
bits of oil burst into flame and it smelled awful - but the chain
looked like new when it was finished. Just be careful not to heat the
metal too much, or you will melt the solder.