In two basic ways: he appeals to Laertes' sense of duty and honor in regards to his father, and, he flatters Laertes. He starts by challenging his loyalties to his father: "Was your father dear to you? Or are you like the painting of a sorrow, a face without a heart?...show yourself your father's son in deed more than in words" (ll. 107-109, 126). Claudius is insinuating that if Laertes really loved his father, he would defend his honor in action. This is a sneaky way to get an honorable man to feel that he is being honorable, by defending his father's death. And, Laertes buys into it.
The king also hints around about Laertes' skill with the sword, and that certainly Hamlet would accept a challenge to a duel, where no one would blame Laertes if Hamlet were to "accidentally" die. A duel is an accepted form of competition; it is a noble way to challenge Hamlet, and the king has suggested that Laertes should certainly take Hamlet-of famed skill at the sword-on in a challenge.
So, Claudius plays on Laertes' grief by challenging his loyalty to his father, and his pride by flattering his fencing skills-all to his advantage.