You can get to Waukegan on the North Shore Bike Path/McClory Bike Path. It's crushed limestone, and starts in Highland Park. To Highland Park you can take the North Shore Channel Trail to the Green Bay Trail (the way I take to work).
The only issue is, from Waukegan, I'm not sure what the most bike friendly route would be. The trail basically runs right along Sheridan and the Metra Union Pacific North Line. So it would be about a 5 mile ride down say Washington. Looks easy enough on paper, but Washington is a main route, so it might not be cycle friendly at all. And it's probably a 35 mile ride just up to Washington.
The Des Plain River trail runs just east of the park.
I suggest you take the McClory/Green Bay/whatever Trail as Chucko mentioned, but turn off onto the North Shore Bike Path at Lake Bluff and head west until you hit the Des Plaines trail, then head north to the park. This will avoid Waukegan, which is a dump. All three trails I mentioned connect directly. Here is a site with more info: