If you must drive, take the Alaska ferry from Bellingham, WA; I would avoid the ALCAN. The ferry is 3 or 4 days, drops you in Haines, AK, and it's a comfortable 2 day drive from there. It's like a low rent cruise. And you could drive your own car instead of renting something (unless you have A LOT of plants)...that part of the trip is pretty well paved. There is a book dedicated to the journey; can't remember the name but I think it's something obvious like "The ALCAN". A google search should do the trick. It's got everything you want to know about the ALCAN; where gas stations/hotels are, what places are open seasonally, road construction, etc.
Or, get new plants when you move, and fly the cat in the cabin with you (after a high strength tranquilizer, of course). I can't imagine the plants are valuable enough to warrant the cost and A$$ pain of driving. Some airlines must allow it, cause I just had a dog and cat on my flight the other day.
When we moved up there in 2001, I drove our truck from OKC to SEA, picked up the hubby and well-tranquilized dogs there (they flew from NC), and caught the ferry in Bellingham. After coming off the ferry, we drove to Tok, spent the night, and made the rest of the drive the next day. This time around (we are moving there after the house sells in Louisville), we probably aren't going to drive anything up there. We'll let UPS take care of everything; and whatever they can't handle, we'll give away.