Unlocking this achievement will also unlock Ratfolk as a playable race.
Do not kill any rats until you marry the Rat Queen! Run away from all of these encounters! Killing rats ruins the option of marrying the Rat Queen.
Recommended to start as a bard, as you'll need music. Otherwise, try to buy the lute from Alluris, and learn music from the Elf in Mythrandir
I'm not sure if it can occur during any standard encounter with rats, but you'll need to play music during an encounter with them to get them to follow you. I know for a fact that it's an option from the tavern quests to clear out the basement of rats, though, so I'd just recommend hanging out at the Alluris Inn trying to trigger that quest.
As an aside, the rat pack is pretty useful for various things throughout the game.
Once you have a rat pack following you head to the Caves. The Rat Queen will offer to marry you here provided you aren't already married and haven't killed any rats. I believe that it happens on level 2, but if it doesn't trigger there you may want to try the other levels.