It could be done, no doubt. But, it would be a full plate of driving each day. According to my log, leaving from Michigan, it was seven days to Whitehorse (now, l was driving a 1990 Defender, so a modern vehicle could probably shave a day off, maybe two)...and then you're probably 6/7hours from Dawson City. So, depending where you're at in Central IL...l'm thinking you're going to be in the same round about time frame. The Dempters is 13 hours to Tuk, so...you're probably not doing that straight through, especially depending on when you leave and length of day or if you're good with driving at night on one of the most desolate roads in NA.
So, best case scenario...it's five days to DC, day and a half to Tuk. Repeat the other direction. I believe this would be a very mentally and phyiscally challenging endeavor. You have no room for error or incident.
Edit #1: Depending on when you go, road construction can be a bugger. I believe on the Dalton, just to the Arctic circle...there were two or three traffic stops that lasted well over twenty/forty minutes each. Heck, one was so long...just about knew the flag girls entire life story....or worse yet...any longer...and up there common law marriage might apply.
Edit #2: Wildlife is a huge concern also. In my travels to Alaska, number of animals l saw killed along the road...three moose, five bison and several smaller mammals. So, if l saw that many...how many made it farther into the bush? Real concern, while traveling...and if a person is trying to "make good time"...that only enhances the chances of an accident. And, you're going there to see wildlife...so driving safely, for your sake and theirs is paramount.
Edit #3: Breakdowns. Got waved down, along the Alcan by a couple looking for a Satelite phone (no cell coverage for about 300 miles of that stretch between Watson Lake and Ft. Nelson) that had been waiting three days for parts to be delivered from Fort Nelson's Auto-Parts Store. And not to knock Canada in the least, it is what it is, and the issues are driven primarily by geographical location (vastness), demand and population. But their supply-chain for parts is not nearly as extensive or readily available as in the states. Just another issue that can develop along the way. The more modern or common your vehicle is could help speed up repairs, certainly. Don't, ever, be in a hurry (or plan on making it home for that matter...lol) if you're driving a 1990 LR Defender. ;)