Frank, on a normal year, June 1-5 might be considered too early because of snow and high water streams. However, this year being a dry year, there won't be so much problem. The top two miles of the Mount Whitney normally has some snow patches in early July, so early June could have some big problems. There are some nooks and crannies on that two miles that don't see much sun, so those are the problem areas. OTOH, those two miles are so heavily traveled that there will be a slippery and muddy foot track across all snow obstacles. About the only place where it gets really tricky is about a half-mile from the summit. There, the trail goes up a short steeper patch, so that would be where the ice axes and ski poles come into play. I would say to go ahead and plan the trip for June 1-5 and plan on taking everything. Then by one week before, you should be able to get updated trail conditions reports, and you can decide whether to delete some of the heavier pieces. The streams between the Kern River and Guitar Lake may be at flood stage, despite the dry year.
If you are taking only those days, then I assume that you are going one-way only (going east), and not returning to Crescent Meadow. That means that you intend to exit at Whitney Portal. There could be some major snow obstacles on that east side as well, like around Switchback 45 on the 97 Switchbacks. That is where the steel cables are, and that is normally where ice builds up on the trail.
I don't think that the wildflowers will be good by June 1-5. On a normal year, the wildflowers up around 14,000 feet peak around the end of July. For this year, I would expect that to be around the beginning of July.