Always check the Yosemite National Park Wilderness Conditions page before you head up -
http://www.nps.gov/yose/planyourvisit/wildcond.htmThey specifically mention current conditions at Sunrise and Cathedral as on June 7: "The Sunrise and Cathedral Lakes areas are 30% snow-covered and the lakes themselves are clear of ice.
The outlet of Tenaya Lake at the Sunrise Lakes Trailhead is an 18-inch wade."
In my experience the rangers typically report a worst case scenario at the time of posting so people do not get into unexpected conditions they aren't prepared for well enough. Yesterday's report says trails are still covered above 9500 feet with more snow on north facing slopes. That hike up from Tioga Rd to Cathedral has an elevation gain from 8600 to 9600 directly up a north facing slope but its only at 30% coverage which is very easy to navigate around.
In your route I would be much more concerned and monitor the conditions up until the day you enter the wilderness for Glen Aulin. With all the snow melting off, the trail below the Glen Aulin HSC is flooded and going to require up to a half mile wade to get upstream which sounds like no fun with a pack on my back. Perhaps heading into Lyell Canyon or up to May Lake might be better alternatives for the last days of your trip.
Rangers are usually very good at posting changing conditions within the valley and along the JMT. However most other areas of the park you will have to call them to get more detailed information. Sometimes they are busy and don't want to talk on the phone with ya for 20 minutes going through their conditions, but often times they are more than willing especially if you call in the afternoon.