The latest I've seen leftover snow is mid-June in Anaktuvuk Pass. I've seen lakes still mostly frozen by July 5 on the Slope. While you could hit a freak snowstorm in July, particularly in the pass, the road is plowed all year and will not be frozen in July. Rain would be your more likely adverse weather encounter, which can turn the road into a mess. As AlcanRider said, the fog can be thick, common, and can drift quite a ways south over the coastal plain.
2007 R1200GS; 2006 DL650 V-Strom; 2011 F800R (sold); 2013 CRF250L (sold); 1995 R1100GS (sold); 2006 Ducati Multistrada 620 dark (sold); 2013 F800GS (woot!)