This is the only explanation I can think of.
A"player" is someone who "plays" the game of love. They fool people into thinking they love them, when in reality they don't really care about the other person in the relationship. The person in the relationship with the player is being "played". Players take pride in "playing" as many different kinds of people as possible, because this shows that they can "play" anyone.
More on players:
http://www.urbandictionary.com/define.php?term=player
http://www.urbandictionary.com/define.php?term=players
A "kink" can be a person who enjoys kinky things. Kinky things, as Elliot pointed out, are fun things in bed... They are basically out of the ordinary sexual practices that are enjoyable to said "kink".
More on kinks:
http://www.urbandictionary.com/define.php?term=kink (ignore the first definition, the rest are more helpful)
So, to "play a kink" is to manipulate someone who happens to be kinky into having feelings for a player.