A chain is what occurs when you create a hand and the cards above it fall to make another hand.
This is the hardest concept to control, let alone master. Thankfully up through the 5x chain you will have only a small amount of trouble.
Learn to use bombs to start off your chains, see moves in advance, and use your slow-down trigger. With practice, you will easily be able to make chains above 5. True masters and Tetris Attack veterans will eventually reach the 24x chain, so don't stress yourself if you find your skills lacking. Just keep practicing and keep the following tips in mind.
--Watch the top chainers' videos and see how they move and try to anticipate their next moves. All of the pros move their cards up toward the top to get a better view of the situation and begin their chains then.
--Use bombs to start off chains when you can. These count as the first hands in chains when done right, which can make the difference when you end with a 6x chain and could have had 7 had you started off with a "bang".
--Shoot for the ground. There's no reason to try and hit 24 your first time. Practice getting 2 and 3x chains in the beginning. Once you find these easy enough, move on up to get the 5x achievement.
--Practice anticipating outcomes. Take a picture of the screen with all the cards moved near the top of the screen. Print off the picture and go through marking your moves, which cards will subsequently drop, and how large of a combo you can get from that one screen.
Above all else: practice. These are the hardest achievements in the game by far. If you find yourself practicing daily and constantly, take a break. Remember that the game is primarily for fun and that Poker Smash isn't the best game for completionists to attempt.
Have fun, and good luck.