How and when do departments decide who is teaching which course?
The department head/chair is the go-to person for such requests. Sometimes there is a curriculum committee that may or may not have some responsibility for seeing that courses are covered. 

When you ask, it would be useful if you go with a plan and some evidence that you are or will be prepared for it. 

You might also talk to someone who has taught it recently. 

Don't wait to make your request. It may take a while if it is unusual for grad students to teach at that level where you are. 

If the faculty is unionized you might run in to a problem. I've been places where the union for adjunct professors might want to complain even though the regular faculty wasn't unionized.