Apple Inc. has set a budget of roughly $1 billion to procure and produce original content over the next year, according to people familiar with the matter—a sign of how serious the maker of iPhones is about making a splash in Hollywood.

Combined with the company’s marketing clout and global reach, the step immediately makes Apple a considerable competitor in a crowded market, where both new and traditional media players are vying for original shows. Apple’s budget is about half of what Time Warner Inc.’s HBO spent on content...