The interview will be scheduled for the next available date, you can't request a certain date.

However, if the date they give is inconvenient, you can request a LATER date via email, and they are good about rescheduling it. But you'll never get an earlier date than the one they give you!
I'm not sure who schedules the interviews, NVC or the Consulate, because the process has changed since we completed our K1 last year. But I would think the foreign Consulate would still do it.