Notice how your professor teaches their classes. If they're more formal lectures, then this person's probably a more formal person and may not be as open to befriending as other professors. If the teacher is less structured and runs discussions, then they may be a little more casual and a little more open to spending time with a student.
Keep the beginning relationship restricted to your fields of expertise. Don't start off asking your professor if they've seen a specific TV show or a movie. Instead, start off asking them questions about their lecture or ask for further clarification about something you read in a textbook or journal.
Let the professor take the relationship to the next personal level, talking about movies, friends and TV shows. Professors are extremely busy people and probably don't have the time to actually become friends with their students. They can be friendly and help students attain their goals, but probably won't be someone you hang out with after graduation.
Make sure you don't take the relationship to a sexual or romantic level. If you feel the professor is interested in you in this way, address the issue or simply back off. If you feel uncomfortable or threatened, go to your professor's supervisor.