I applied through a recruiter. The process took 4 weeks. I interviewed at US Navy (New York, NY) in Jan 2015
Interview
There was no interview process. You sit at MEPS all day to hurry up and wait, and then ship off to bootcamp. The closest to an interview process is when the MEPS recruiters ask very intrusive questions and prod into every significant and insignificant detail of your life to ensure you're not a russian or chinese spy.
You have to do a background check, several physical fitness test, past a background check, eye and ear check. Full medical overview. Polygraph. No outstanding warrants. Clean background or waiver. Certain tattoos. Boot amp is several months then you are assigned an A school to attend. Pass that and your off to the fleet
You walk in. Sit down. Take the ASVAB so that recruiters can get a good idea of how stupid you are. Once they get a score from you they will have options that you would fit well into based on the needs of the Navy. Tell them your interests but they ultimately will put you where they need you.