First of all there are two things you need to fulfill:
a) There is no authenticator attached to your account and never has been
b) You need a friend or someone you trust who got an authenticator
c) optional: voice chat is very helpful here for method b (if you are paranoid or don't really trust your friend)
Method A (you trust your friend):
1) Give you account details to your friend
2) He logs into you battle.net account
3) He attaches his authenticator to your account
4) Log in, loot core hound
6) Let your friend detach his authenticator from your account
Method B (you are paranoid/don't really trust your friend)
1) Log into your battle.net account
2) Add authenticator
3) Tell your friend in voice chat/messenger to send you his auth key
4) If you entered the key, tell him to send you a message with a generated code (needed to activate)
5) Log in, loot core hound
6) Detach the authenticator, you need two following generated codes to do that
Be aware that you cannot use your core hound after the authenticator was detached. You cannot summon him anymore. But it counts for the achievement as the flying machines and carpets do for the mounts achievement although you gave up on engineering/tailoring. I hope it helps some to get this one.