You're mixing systems.  The train station in Palatine is part of the
Metra system which runs trains from the outer suburbs to downtown. See:
http://www.metrarail.com/

The L or Elevated system is part of the Chicago Transit Authority.
http://www.transitchicago.com/
The NW EL line only runs out to O'Hare airport.
Metra and the CTA are separate systems.

You might try Illinois Beach State Park.  It's pretty far north of the
city, but there isn't a lot of camping in the city.  See:
http://dnr.state.il.us/lands/Landmgt/PARKS/R2/ILBEACH.HTM
You could probably catch the Metra in Waukegan or maybe Zion see:
http://metrarail.com/Sched/cnw_n/cnwn.shtml
I suspect that you will have to pay to park at the station.  A lot of
areas require permits to park on the streets near the stations and you
will be ticketed if you don't have a permit unless you park in a metered
space.  But parking should be cheaper than parking downtown.  You might
want to check with the Park to see if you can catch a bus from the park
to station.

There are a couple of private campgrounds in the southwest suburbs.  I'm
not familiar with them but you might find them on google.

There is also Martin CG just a few miles west of Windy City. I've stayed
there twice and prefer it to Windy CIty.