Absolutely! In fact, a lot of people do that. I have a ton of advice, of course, and I will write you a much more detailed response when I'm not so delirious. You really can also do a day cruise to get a sense of the cruise scenery (which is unlike what you'll see driving) without having to be stuck on a boat all day. There are day cruises out of Seward and also the famous 26 Glacier cruise out of Whittier, close to Anchorage.
From Anchorage, you'd want to drive both south to the Kenai Peninsula (Kenai, Seward, Homer at the tip of the spit), and then north to Talkeetna and Denali. That's the most common route and I'd suggest two weeks for the whole thing. The distances here are more than people anticipate, and there's usually only one possible way to drive anywhere (i.e. no alternate routes). The scenery along the drives is a major part of what there is to see. Will you be camping, RVing or staying in hotels/B&Bs? ETA: Oops, I see you said you wouldn't be camping. RVs are a good way to go (and you can sleep any time!), but crazy expensive between rent and gas. There are lots of small places to stay depending on what your route is, but summers do book up. Again, let me know what you're interested in and I can give you more specific pointers.
There are lots of other fascinating places to go, like Sitka, Kodiak island, and Juneau, but those all require flights of some kind.
The other thing I recommend to the adventurous is the REI tours, if you want to get a taste of the real outdoor stuff. A college friend of mine is coming here to do one - they're going to go white-water rafting, kayaking, hiking and do some glacier climbing stuff. Even if you're not that extreme, Anchorage has some great hikes close by that are well worth checking out (and depending when you're here, I'd be happy to take you, of course!).
If you're into fishing and hunting, that's a whole different world too.
Let me know - I'd be thrilled to give you more specific tips and guidance!