I never had an achilles, but I have had some success with a powder blue tang, so maybe my experience can relate. I think this can really apply to any fish, its all pretty obvious stuff, but not everyone does it.
Have plenty of live rock, and let the tank get established. My tank was about 18 months old when I added my PBT.
Have good water quality. Invest in an RO filter if you dont have one. Also make sure you have a good protein skimmer. I battled bad water quality for months before I bit the bullet and bought the right equipment.
Provide a good diet. I feed spectrum pellets, seaweed, mysis shrimp, and frozen tang formula. I also feed macro alge that I grow in my fuge. I also feed fresh shrimp and squid from the grocery store. My PBT seems to like the spectrum pellets and seaweed the most.
When picking the fish ask to see him feed. Make sure you get one that is eating. I know in the case of powder blue tangs they starve to death in captivity. Also make sure the fish has good color, fins, is active.... obvious stuff.
Finally when bringing the fish home try to make the transition smooth and stress free. I rearranged the rocks in my tank to dsestroy the territories. I turned off the lights. I drip acclimated the fish for about 3 hours, then put him in.
I'm not claiming this is the only recipe for success, but it worked for me.