As a ten year former resident of Prince George who's done most of what you're thinking of I thought I'd throw my two cents in.
I did the trip by myself about four years ago. I went the first two weeks of August. There was already snow on the Dempster Highway. This may not be normal, but it quickly put a stop to my plans to go up to Dawson City as I didn't have my snow tires on or chains I could use in an emergency. I decided to continue on to Whitehorse and spend more time camping and hiking in Kluane National Park, a truly special place.
I was driving a 2001 Subaru Forester (still do) which is all-wheel-drive. It came in handy a few times as some of the 'highways' way up in northwestern BC were barely passable mud tracks. If it has rained they will be slick and the safety barriers you're used to on highways don't exist. South of Dease Lake and North of Bell II the road is dodgy. I don't know if the province has done anything about it since I went over it, but it was very hairy in parts. There were many kilometres I could barely manage 20km/h because the roads were so rough. Gravel the size of your fist. Yes, that was on Highway 37.
Depending on when you go, be prepared to travel loooong stretches without seeing anyone. That took some getting used to. If it's peak tourist season with lots of folks driving the Alaska Highway, you'll see lots. Like I said, I went the first two weeks of August and more than a few times went hours without seeing another vehicle, particularly on Highway 37. I can't tell you how happy I was to reach Highway 16 at Kitwanga.
Bring as many emergency kits as you can fit. Not only did I have a spare tire, I also had tire patch kits and and air compressor to fill the tires. I'm a forestry guy by training though those days are long behind me, but I was comfortable going over rough roads alone, days out of reach from cell coverage and many hours drive from civilization. It can be both terrifying and liberating. Have a jerry can full of gasoline. Just because a community says they have a gas station doesn't mean they have gas to sell you or will be open when you get there. Have a tool kit that you know how to use. Have food and water. If possible, bring a GPS so you know exactly where you are and/or the SPOT system so you can signal for help if you need to. Make sure your maps are up-to-date and you can read them while driving.
Having said all that though I had no trouble whatsoever. Didn't even pop a tire, something I had done many times before while exploring the backcountry. I never ran out of gas. But, my tires were shredded by the time I got back to PG. Had to replace all four as the sidewalls were sliced. That shouldn't stop you though. It's the almost endless stretches of wilderness BC has that I miss most about the province.
You will see some of the most spectacular landscapes. The drive in to Stewart BC will take your breath away, quite possible the most beautiful and awe-inspiring 50kms I ever travelled. Glaciers you can just about reach out and touch. The Yukon is everything you imagine it will be and more. The people are fantastic. There are wild Buffalo to be seen wandering the highway north of Liard River, BC. Make sure you stop in at the Liard River Hot Springs and camp the night in the provincial park and take a dip, it'll warm your soul. There were wild horses blocking the highway south of Dease Lake on Highway 37. I didn't mind stopping though as the surroundings were almost too much to take.
If you have any questions let me know and I'll do my best to answer them. It was two weeks of my life I will never, ever forget. The Yukon alone will make you cry.
Go. And bring as much film as you can afford.