Hi there:
You will have to use JFK's airtrain from JKF to Howard Beach Station. Then take the subway from JFK to Manhattan Penn Station, then take PATH train to Newark's Penn Station.
To get to Teaneck you will have to ride NJ transit. Check the websites for bus routes, schedules and fares:
mta.com, njtransit.com, panjny.gov/path
Once you get to Teaneck you will have to get a taxi, unless you know someone.
I estimate your combined fare to be around $20 or less. This is the best way, cheaper than a taxi.
It's not easy to move around New Jersey my friend. Good luck. :P
NJ Transit