Agreed that $30-$40 will be difficult, not impossible, but difficult. Like fti mentioned, you'll want to spend as much time in the backcountry as possible so you're not paying for campsites, but that comes with its own costs like freeze-dried food, water purifiers, etc. If you already have all the gear you'll be fine, but it can add up quickly if you don't. If you're experienced with backcountry camping that actually sounds like it would be a wonderful trip. You could also spend some time in Denali National Park, then its lesser-traveled and lesser known sister Denali State Park, and then hike in the Chugach National Forest near Anchorage, there's no shortage of excellent hiking/camping/backpacking in AK.
Two things to look into are the Northern Lights and Alaska Toursaver coupon books. Both of them cost money (I forget how much) but they offer 2-for-1 on the Alaska railroad, rental car discounts, and discounted campsites, iirc. You might want to check it out to see if the savings > cost. After beginning of July you can often buy them for half-price on eBay.