In episode 3.2, Martha mentions that he had two children, Boomer and Lottie.
Martha moves in with him in episode 2.17 (Tick, Tick, Tick. – (Part 1)), but as of episode 3.2, Martha is considering breaking off their relationship as she feels the fire has burned out. Before she can talk to Chet about it, he proposes, and she promises to think about his proposal, only to find next day that he died of a stroke the previous night.
After Chet dies, in Lucky Stiff, Martha inherits $1 million from his estate and after agonizing over whether or not to accept the money, Detective Kate Beckett helps convince her to keep it. Martha eventually decides to use the money to open an acting school, the Martha Rodgers School, in which she promises to hang a portrait of Chet in his memory.