What is the weather going to be where you live on the 3rd of July, 2013? If you can accurately predict that, we might also be able to tell you what the weather will be in Deadhorse at any given time in the future.
Normally, by the first week of July the snow will have been gone for several weeks. But it could also snow there on the 4th of July. When you're that far north, and in a place that has weather that is influenced by such a variety of factors, you'd better be ready for anything. In addition to the possibility of snowfall in Deadhorse, it's more likely that you could encounter snow over Atigun Pass or pea soup fog for many miles south of Deadhorse.
Considering that late May - early June is usually some of the best time to ride to Deadhorse, your planned schedule should work out well. But nothing is certain on the Dalton.