Here is the full list of all 45 DPS IDLE achievements.
Get 25 gold cards
Get 60 silver card
Get 5000000000000000 poisonous DPS
Enhance the sphere of magic to 31
Kill 10000 monsters
Get to the second location
Hire two heroes
Get 5000 leafs
Boost the card to 7
Get 3 gold cards
Get 30 silver card
Get 500000000000000000 DPS
Get 5000000000000000000000000 gold
Hire one hero
Get 1000000000 poisonous DPS
Boost the card to 4
Get 1000000000 DPS
Get 100 leafs
Get 100000000000 gold
Join the guild
Kill 500 monsters
Get 5000 poisonous DPS
Enhance the sphere of magic to 6
Get a silver card
Boost the card to 2
Get 10 leafs
Get 300 DPS
Get 5000 gold
Open the sphere of magic
Kill 10 monsters
Join the 3 guilds
Join the 5 guilds
Get 50 gold cards
Hire 5 heroes
Get to the 4 location
Get to the 8 location
Kill 2 bosses
Kill 5 bosses
Kill 15 bosses
Open up a new continent
Open up to 5 continents
Open up to 24 continents
Boil potion
Boil 5 potions
Boil 20 potions