Depends what you can/are willing to tolerate. I twice started from Seattle in late May. Took three days to get up to Bay View, which is just east of Anacortes. Crossed the North Cascades Highway (Rainly and Washington Passes) three days later. It started raining at some point during the climb. The rain turned to snow approaching both summits. The upside was that the scenery was dramatic and there was very little traffic. We also had snow in Republic and then climbing Sherman Pass several days later. Nothing stuck to the roads.
As noted, there are timning issues with Going to the Sun Raod, which you should realistically not expect to open before at least mid-June unless you get there on a weekend, the road has been fully plowed and there are no hiker/biker restrictions on the road. The current road rehab project also affects the opening date. From Bay View I took exactly 14 days to get to GTS, including two rest days. That was not fast pace, but in some cases it was dictated by the spacing of camping and other services. You really don't want to miss GTS. The optional way around is a long drag with scenery that simply does not compare. It also leaves you with more miles if you want to get back on route to do the section into Canada, with a stop at Waterton Villeage, which I highly recommend.
East of the Cascades can be roasting during summer. Both times it was very warm from Winthrop, WA to Tonasket, WA in early June. I would not want ride through that shadeless stretch during a hot day in July. I have read and heard stories of people riding west towards the Cascades in brutal heat.
If I were to do it again, I would probably start from Seattle around June 1st and be prepared to hang out in Glacier if necessary.