Each of these items are called "Bounties". To win each bounty you must complete 3 challenges in the game. The challenges differ from Going "All-In" in a hand, to winning a Showdown with 3 of a Kind or Better. Once you have completed each of the 3 challenges, you must win your next tournament to unlock one of the bounties that each of the individual characters put up.
Note: The Bounty challenges will not be available until after your first tournament. To see what your current challenges are, press or access the Bounties Screen in the Pause Menu.