Here is the full list of all 10 Four Kings: Video Poker trophies - there are 7 bronze, 1 silver and 2 gold.
Win your first hand
Lose five hands in a row
Run out of your first set of chips
Keep your initial draw and win the hand
Get a Three of a Kind with Aces
Win 100 Hands of Video Poker
Win 200 Hands of Video Poker
Play 500 Hands of Video Poker
Get a Full House
Get a Four of a Kind or better