One July plan has me going into Woodchuck Country, Hell-For-Sure Pass, and Muir Pass, if you know where those are. Unfortunately, by the 3rd or 4th week of June I am likely to be in another state.
For Yosemite, the NPS site has trail conditions, but those are notoriously slow to be updated. I watch the Tioga Road snowplow progress and try to guess things from there. Also, you can find the reports of the winter rangers at Tuolumne Meadows and guess from there. There is a lot of pressure by the Mono County businesses to get the Tioga Road opened up as early as possible, but that costs money. Even when the road gets opened, that does not mean that the high country trails are passable. Even when they get passable, that doesn't mean that the country will look good enough to warrant a backpack trip. The California Water Resources site is concerned about snow depths and water content, but that doesn't say much about the readiness of a trail.