There are six total playable characters. Each time you unlock a new character, I'd recommend rerunning the 1st dungeon (Caeldippo Caves, Silk Caverns) using the new character since I find it has the easiest boss.
This also ensures you don't have to keep track of which characters have beaten bosses, and which haven't.
I highly recommend having a couple points in Armor Up and Warfare (Workshop) to make your runs easier.
Achievement tracking DID work for me.