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$30-$40/day total will be next to Impossible. But spending as much time in Denali National Park as possible will be key. Definitely camp at either Tekanika or better yet Wonder Lake Campground. That will give you the chance to purchase a camper bus ticket. That costs about $40 per person for as long as you stay inside the park. Once you come further east than mile 20 of the Park Road you need to buy a new ticket. Another option is to go back country camping for part of the time. There is no charge for Backcountry camping and the camper bus ticket is valid for that as well.

Buy all of your food 4 Denali in Fairbanks. It is much cheaper and you get much more variety. If you stay inside the park for 2 weeks and Backcountry camp, you can cache some of your food in food lockers at various places inside the park. There is some great hiking in Denali National Park, both day hiking and overnight hikes. Ask a park ranger, Rangers in the backcountry office, or the bus drivers. There is also a book you could purchase in Fairbanks by Ike Waits with something like 40 suggestions for day hikes or overnight hikes in Denali National Park. A great place to stay at the beginning or end of your time in Denali National Park is the Denali hostel. Excellent, clean and very friendly. They have transportation several times a day to and from the park.

It is true that renting bicycles can be expensive. I have purchased more than one bicycle in Alaska. Inexpensive bicycles are less than $100 each. Bring them with you and sell or give them away at the end of your trip.

Taking the bus from Fairbanks to Denali is probably the least expensive. Then from Denali to Anchorage there are at least two bus companies or the train. I am not sure I would stay in Anchorage for too long. There are some interesting things to see and do for several days. You might consider renting a car for a couple of days to get out of Anchorage. The area south of Anchorage for an hour or so is a jackpot of activities that are free or very inexpensive. If you can take your bicycle from Fairbanks to Denali to Anchorage on the bus or train maybe you could stay in Girdwood and use your bicycle to get around that area. The train and at least one company, Magic Bus, can also get you between Anchorage and Girdwood. I am pretty sure there is a second company that does shuttle service between Anchorage and Girdwood but the name fails me at the moment.

If you don't have enough points for your entire time, there are some decent hostels in Anchorage. There is also a public transportation system in Anchorage called The People Mover. It can be relatively slow with all connections going through downtown but it is Fairly reliable and inexpensive.

FYI when I tent camp in Alaska I figure $10 to $12 per day for all my food. That is eating very well and cooking all of my meals. You can get by cheaper if you buy things like mac and cheese Etc. Stay away from the freeze-dried food since they are very expensive.

If you take the bus from Denali to Anchorage you might consider a couple of nights in Talkeetna. There is a campground right on the water and the bus company charges $10 more for the Stop in Talkeetna than they charge for the trip from Denali to Anchorage without stopping.