In agreement with the above responses. Late August to early September is ideal. Weather changes quickly just as the short season comes and goes. Also, little to no bugs (and people) later in the season. Here's my experience:

11 Sept 2016
Guitar Lake to Mt. Whitney (JMT thru-hike)
Day prior missed the heavy rains by approximately 1 mile distance. Began at 2:00 am, cold but bearable, clear skies, summit before sunrise, clear skies all morning, no wind, no significant snow on trail, cloud coverage by mid-day, dry trail to Whitney Portal, moderate rain at Whitney Portal by 3:00 pm. Not too crowded.

1 Oct 2017
Guitar Lake to Mt. Whitney (Horseshoe Meadows)
Day prior dry and cold. Began at 2:00 am, very cold and unbearable, summit before sunrise, clear skies all morning, windy, no significant snow on trail, partly cloudy by mid-day, dry trail to Whitney Portal, warm and dry at Whitney Portal by 2:00 pm. Very little crowds.

21 Jul 2018
Guitar Lake to Mt. Whitney (JMT thru-hike)
Day prior pounded by heavy rains, hail, and lightning storm. Began at 2:00 am, cold but bearable, clear skies, summit before sunrise, clear skies all morning, calm wind, no significant snow on trail, cloud coverage by mid-day, trail to Whitney Portal flooded at several locations, difficult to pass, partly cloudy with chance of rain at Whitney Portal by 3:00 pm. Very crowded.