It'll really help if you can provide a few more critical details: type of plane (fuel range, est. cruise speed, floats/wheels, 100LL or JetA), time of year leaving (what month window, if nothing else), expected accommodation (camping or hotels?), and time window available.
See, if you're ferrying a KingAir with non-functional icing system, I'll give you a lot different advice than if you're in a supercub.
In general, get your passport now, if you don't already have it. If you can find an out-of-print book called the Alaska Airmen's Logbook, it's the best guide I've ever seen to routes within Alaska and down to the Lower 48. Look up customs info, and remember that customs info changes all the time, so anything I could say would probably be outdated by the time you get there.
There are two major routes: the coast and the Alaskan-Canadian Highway, or Alcan for short. I do not recommend the coast for anyone who doesn't have speed, range, deicing, and IFR... or floats, long range and ferry tanks, and a lot of time on their hands. The Alcan is longer, but takes you on the right side of the Rockies for your eventual destination, and not only are there regular fuel stops, you have a long landmark and emergency landing strip underneath you.
There are several variations on the Alcan. We don't need to worry about Dease Lake, as you're not aiming toward the West Coast. Unless you have something with lots of range (or floats) and darned good weather, I won't recommend the Trench (a shortcut consisting of a long lake-filled trench between the mountains, with uncertain fuel availability, no weather reporting, and very few landing spots). As you look at the map of the highways, though, I will recommend leaving the Alcan proper before it winds down to Vancouver BC and Seattle WA, and heading on the highway over to Cut Bank or Billings, Montana. The latter two airports are used by many pilots to clear customs when heading north in the spring and south in the fall, so customs there is both used to GA and even, as much as customs can be, GA-friendly. Polite, at least.
A word on maps: if you know someone who's run the Alcan recently, their maps are probably still current. Canadian maps do not expire, like US ones - they supercede, with sucessive printings only being done when changes require it. This means you'll want to check your maps against the most current to make sure they're still good, but you can save a pretty penny this way (canadian VNCs - equivalent to American sectionals - are pricey). NavCanada, as of the last time I looked at the route last fall, had an official statement out that they were disconinuing their version of WAC's for the areas that included the Alcan. I don't know why. On the other hand, with the uncertain publishing date of the VNC's, this means it is much more critical for you to have their A/FD equivalent, which does expire every 56 days on the same schedule as USA ones. Northern Lights Avionics in Anchorage has all these in stock, and can mail them to you or you can pick 'em up at Merrill Field when you're up there before you go.
A word on weather: Weather reporting is spotty at best, and even if it is accurate for the area reported, there are no guarantees that the weather in a pass or on the other side of a pass are going to be the same. The Alaskan FSS and weather briefers are not contract, are local, and really know their stuff. Call them and talk to them, and they can give you much better advice than a computerized forecasted guess, or someone staring at a computer in California reading off the weather for a pilot halfway across the country. Once you get to Canada, flight plans are mandatory - but the Canadian weather briefers and NavCanada guys are used to Americans running the Alcan, and can give you plenty of useful advice and information if you ask politely.
Also, June is the best month in Anchorage - least rainy days. Plan on weather delays, and bring a book you've been wanting to read along. You may end up in a tiny town with nothing to do for a week (especially in winter), and it is BAD to let boredom push you into making decisions that are fundamentally risky. That said, you're more likely to have a straight shot down the Alcan with no delays than you are coming up, especially if you start down on the heels of a front, chasing it in the clear air behind it, instead of coming against the flow of the weather.
A word on internet: do not assume it's everywhere. In fact, I don't know of anywhere in Whitehorse you can get wireless to file the EFIS to cross the border - most guys I know file with a delay built in here in Anchorage or in Fairbanks, then take off in a heck of a hurry toward the border to try to make it in time.
That enough advice yet?