Here is the full list of all 10 Four Kings: Video Poker achievements.
Get a Four of a Kind or better
Run out of your first set of chips
Play 500 Hands of Video Poker
Win 200 Hands of Video Poker
Win 100 Hands of Video Poker
Get a Three of a Kind with Aces
Keep your initial draw and win the hand
Get a Full House
Lose five hands in a row
Win your first hand