You are greatly underestimating that cost
The way to do it cheaply is to fly in late June/July and camp in Denali park. The weather is warm enough for camping and its fantastic. I actually feel camping is the best way to do it!
Assuming you already own camping gear. And can pack some food.
Here's how it can be done:
$100 - Camping in Denali Park: $45 (entrance fee) + $17/night x 3
$200 - $50/day * 4 days car rental
$50 - Fuel / Snacks ( 300 miles round trip from Anchorage)
$50 - Denali Bus Ticket
$50 - Firewood/Misc
_____
$450 - 4 days/3 nights.
Budget left for flight $550 .
Depending where you fly in from you could come in under 1k. Seattle round trip flight in June $400. NY/Alaska is $620 round trip. Which would push you over 1k. But if you convince a friend to go with you, you could split the car/camping costs.
You can also do wild camping for free, and if you're into trying fly fishing, I definitely recommend getting a fishing license.
Fishing License Costs: 1 day - $25 / 3 days - $45. A basic Fly fishing kit can be bought < $100.
1- Live close to Alaska.
I could try driving there, but the Bering strait might be an obstacle.
1. Fly out to Anchorage, Alaska
2. Drive to Denali visitor center, (Denali State Park)
3. Take the Bus to Denali (~$50)
You are almost guaranteed to see a wild grizzlies along the way. The bus will stop so you can take a photo. You can even get off and go for a hike if you like. They will pick you up on the way back. (or whats left of you...) If you decide not to, thats okay too. You will have many near death experiences, as your school bus drives along a dirt road along steeped cliff mountain edges, for many miles. The bus takes you to heart of Denali, where you can see Mount Mckinley (Highest mountain in North America), and where temperatures can dip to minus 60 degrees in winter.