Fly to Anchorage and rent an RV. Plan a circle route that takes in Fairbanks, Valdez, Seward and Homer. Consider the ferry between Whittier and Valdez. Two to three weeks would allow time for driving the circle and time to stay and do something in a number of places.
Just fuel to make the trip from FL to AK and return will run in the range of $5,000. An RV rents for around $125/day plus fuel and such.
Check with a cruise line. Take a cruise to or from AK one way, do your own tour of AK or use the cruise line's tour, fly back to FL.
A drive tour would need more in the range of 2 to 3 months, unless you plan to drive 12 hours a day with minimal stops. If you travel 6 to 8 hours a day and cover 350 miles and stop once a week for laundry and such, and don't do any sightseeing enroute, you will need 2 weeks+ each way just to drive.
Best wishes for a good trip.