Talk casually with your teacher. Share jokes, talk about popular movies or discuss a political event. Your conversation does not need to revolve entirely around your academic studies, and talking about other things with your teacher before or after class can help you learn a little bit more about her as a person.
Focus on the primary goal of your relationship, which is to learn from your instructor. If you find that your relationship with your teacher extends to outside the classroom and has become more friendship based, your relationship may have expanded too far. A close friendship outside the classroom with your teacher could destroy your objectivity inside the classroom. While it's fine to enjoy your teacher's company, it's also important to keep your relationship professional.
Avoid sucking up to your teacher. Nobody likes the student who tries to be teacher's pet. While you may thoroughly enjoy your class, notice your classmates and demonstrate respect for them by sharing your teacher's attention. For example, don't jump to answer every question. Allow your classmates to answer a few. Try not to dominate classroom discussions even if the issue is one about which you care deeply. After class, give other students a chance to talk with your teacher before you start monopolizing his attention.
Show up on time to class and be ready to work, demonstrating that you care through promptness and being prepared. Don't turn in work late and hope your teacher lets it slide, as this situation can put your teacher in a difficult position and can damage your blossoming relationship. If you struggle with a specific project or theory, talk to your teacher after class to see if she can help you or if she can refer you to someone else who can help.