NEW YORK—Fox News Media is getting into another rough-and-tumble world—the weather. The company is launching Fox Weather, a free streaming service and app that it expects will be used most frequently by customers on their mobile devices. In preparation for the Monday start, Fox has been building staff, plucking personnel from The Weather Channel and markets in New York, Houston, Phoenix, Miami, Charlotte, North Carolina, and elsewhere. Advances in weather technology and interest created powerful storms has made weather news a popular area in which to invest, experts say. “It’s all we talk about,” said Sharri Berg, Fox Weather’s president. “We might as well build a platform for it.” Fox Weather is moving into a market dominated by long-time players like The Weather Channel and AccuWeather—both of whom have been busy building their own new products—as well as aggressive niche players. When Berg scrolls through weather apps, she said she …