I've gone most trips during end of may first week of june -

weather can vary from cold and miserable to way too hot.  It snowed 2 to 4" one Memorial Day weekend at Dawson Campground in Quetico

Bugs are just as unpredictable, but generally are just starting to get bad in my opinion

key to dealing with bugs at camp is to select a campsite that is on an exposed point where you can get a breeze from 3 directions - that helps a lot to keep the bugs away during the day - doesn't matter so much where you pitch the tent, just so that you have that exposure for cooking and sitting around time;  if possible, select that point away from small streams - black flies hatch in the streams