In 2005, 8 of us rode from Fairbanks to Vancouver. The details are at http://www.windrath.info/alcan2005/index.html
The Milepost is excellent and it is what we used for our planning - wonderfully detailed although some had gone out of business. You will be on the Al-Can highway for the first part. At Watson Lake you will have a choice to take the Cassiar Highway instead of staying on the Al-Can. It is less traffic with a few less services, but awesome scenery. It is a more direct route than going all the way over to Missoula.
Of course, if you go over to Missoula, you will also see some great areas by Banff and Jasper and the Icefields.
The nice part is there are not many options for roads between Anchorage and the States. You won't need your GPS.
The bugs are terrible. The campgrounds are set up for RVs, so your tent is often going to be on hard gravel. The roads are rough, so your speed will be affected by 2-3 mph. There are not any huge climbs, but the roads are not flat, so you are going up and down all day long. We left Fairbanks at the end of Memorial Day weekend and got to Vancouver right around July 4th. During the first week, we had overnight temps in the mid-20s, so a little chilly rolling out of bed.
Headwinds were quite strong. We were told by the locals that sunny weather means headwinds (going north to south) and rainy weather usually means tail winds. Take your pick.
We only had 3 days of rain.
In June, you will not have a lot of traffic on the Al-Can since most vehicles are going north. As you get down towards Prince George, you will encounter significant logging traffic going very fast. Just have to be a little careful.
Do not think about going south to north because the traffic will be more challenging and, what climbs there are (around Whistler and Lillooet will be much steeper.
Good Luck - it is a great route to ride. If you have questions, let me know.