The answers to the questions you are asking are not clearly given in the book. The author tells us that Kim is from Vietnam, but he does not tell us specifically why she and her family came to Cleveland, Ohio. We do know that Kim has come to America fairly recently; she is nine years old, and has memories of the weather in Vietnam; she says, "an icy wind teetered trash cans and turned my cheeks to marble...in Vietnam we had no weather like that." The story takes place after 1980 - Amir, later in the narrative, says that he "came to the United States in 1980," so Kim would not have emigrated during the Vietnam War.
The cause of Kim's father's death is not given either; Kim says only that she had "been born eight months after he'd died." The author tells us that Kim's father "had been a farmer...all his life in Vietnam," but he gives us no specifics on his death. Also, no details are provided about the manner in which Kim came to America. We do know that she lives with her mother and at least one older sister, but we are not told whether she came here with them, or not.
The author's central theme is the garden, and the sense of community that develops among the people who gather there. He gives us only enough background information about the characters to emphasize their uniqueness and diversity; for his purposes, that is enough. It is left to us to fill in the blanks; to conjecture about the details of Kim's and the other characters' lives, if we are so drawn.