Here is the full list of all 32 Hero Rush: Mad King achievements.
Capture 10 cities
Capture 25 cities
Capture 50 cities
Capture 100 cities
Capture 150 cities
Capture 250 cities
Gather 3 heroes in a team
Gather 4 heroes in a team
Gather 5 heroes in a team
Gather 6 heroes in a team
Gather 7 heroes in a team
Gather 8 heroes in a team
Craft an item level 3
Craft an item level 4
Craft an item level 6
Craft an item level 8
Craft an item level 10
Form a team of 3 archers
Form a team of 3 magicians
Form a team of 3 swordsmen
Level up 3 heroes to level 5
Level up 3 heroes to level 10
Level up 3 heroes to level 20
Level up 3 heroes to level 50
Move to new island
Reach level 5
Reach level 10
Reach level 20
Reach level 30
Build 10 markets
Build 10 jewelers
Build 10 libraries