The item related achievements like Terrarian Sword should work in a similar way, having to find the memory bit that says if you have the item in your inventory. This will also be used on the Monster Hunter achievement, which should keep count of which boss monsters you've killed (21 including event bosses). Upon finding this we'll have to find something that says he got killed, which will be the trigger for the achievement.
That said, for the achievement i want we'll have to hope that there's a part in the memory that says if a specific boss monster is spawned, in this case King Slime. If it's wrong or not valid please hit me up and I'll do my best to get it right i have no idea how to write an achievement logic, but I've read some past submissions for this challenge and will try.