I've done large parts of during a Vancouver to Boston tour. How long? Divide 4200 miles by your daily mileage. Over the length of a USA coast to coast I've averaged around 55 miles per day including rest days. 60mpd for the riding days. Some are faster others slower. So if I was doing the transam I'd allow 76 days plus and extra 3 days or so as a margin for error and sourcing bike packing stuff at the end. Call it 80 days.
It's worth remembering that 90 days is the limit for the USA under the visa waiver scheme. Any longer and you need to apply for a visa and go to the embassy in London for interview.
A west to east tour is best done later. Some of the passes are not clear until June.
http://www.oregon.gov/ODOT/HWY/REGION4/ ... oject.aspx
In 2014, this means the Highway will open to motor vehicles no earlier than Monday, June 16. Prior to then, cyclists and pedestrians may use the Highway following normal rules of the road, and at their own risk.
So starting any time from early June to mid July. I started in Vancouver in July 1st and finished in Boston on 19th Sept. Good weather all the way across. The days were noticeably shorter by then. Only a couple of wet days and a few thunderstorms here and there. A couple o frosty morning in the mountains.