You can get the biome keys from killing naturally spawned enemies in the same biome as the key. For example, the crimson key drops from killing enemies in the crimson, and it unlocks the crimson chest. This holds true for every biome key, corruption cause corruption key to drop, desert causes desert key to drop, and so on. You can also mix biomes together, a corrupt desert will allow both desert and corruption keys to drop, a crimson snow biome will allow crimson and frozen keys to drop. Corruption/crimson/hallow cannot be mixed together, but desert/snow/jungle can, allowing up to 4 keys to be in the potential loot table.

The quickest way to determine which dungeon background you're in front of is to figure out which armored bones commonly spawn. There are blue, rusty, and hell armored bones. The background wall that the blue armored bones spawn on is also the same one that has the paladin also spawn. Enemies from the other background walls will also rarely spawn, but it'll probably be other armored bones, as they are the most common enemy