min of two weeks.   There are some unpaved roads that RVs and rental cars are not allowed on, like the Denali Hwy, and through Hatcher Pass, so just make sure you are aware of any limitations.  A big difference that I see from AK to here is in Alaska, there are many places to just pull off and stop, or camp the night.  No services, but quiet parking.   Lots of State campgrounds, but plan ahead, as they are busy and get full during the salmon runs.  
On what to see, depends upon what you like.  It is a BIG state, and you can do a lot of driving.   Anchorage to Home is a about 7 hours.  Anchorage to Kenai about 3.5.   Anchorage to Talkeetna is just over 2, and Anchorage to Denali is 4ish.  
If you want to fish, head south to Soldotna, Kenai or Homer.  If you like mountains, head north to Denali.    I don't think it is worth the drive to Fairbanks.  The good thing about going to Fairbanks is it gets you closer to Chena Hot Springs.   That is worth the drive.   Seward is a gorgeous place and so is Valdez, if you can visit both when it isn't raining.  Take one of the boat tours out of Seward for the day, like Kenai Fjords.   Pick the tour operator with the catamaran, as it is much smoother to ride on.  You don't feel the wave swells as much as the single hull. 
Don't plan on seeing the whole state.   Too big.   If you really want to see the state, you need to get off the road system and fly out to a village.
I think mid-July to mid-August is the best weather.   Late August into September gets rainy.   Most Alaska rain is soft mist so you can do things outside, just will get wet.   Along the water, they seldom have thunderstorms.   Those are inland, away from the ocean waters.