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