Early June might be OK for Kings Peak, but it is doubtful. Here is a link to Utah weather stations showing snow totals.http://www.wrh.noaa.gov/slc/datamap/ind ... e=0&fire=0
Currently, the ones closest to Kings Peak are reporting from 128% to 180% of normal. Spring snow in the Uintas is typically miserable stuff where even with snowshoes on you can sink in to your knees unless it is frozen solid.
The Wasatch however, typically has great snow that is nicely consolidated in late May/early June and you should be able to get in some pleasant snow climbs.