It's looking like a low snow year, so you may not encounter much ice or snow, even in June.
Your bigger issue is obtaining the permits. To go up the North Fork Lone Pine Creek and spend the night, you need an overnight wilderness permit. For that route, there are only 6 per day available exactly 6 months in advance of your start date. Probably already reserved, on recreation.gov. There are sometimes no-shows, that can be had at 10 AM to start up the same day. Also 4 slots are available the day before -- there's a drawing of numbers out of a hat to see who gets to the counter first at the Visitor Center a mile south of Lone Pine. Those may be your best chance. (not sure what time that drawing is, you should call to find out).
And then it gets worse... If you DO summit, and then exit via the Main Trail, you also need a Trail Crest Exit Permit. I am sure they are already gobbled up, since people coming from trailheads farther away get 7-days advance choice on them. There are 25 slots of the exit permits, 40% held back, but again, people coming from farther north get those first, too. Really sorry to bear such bad news.
Using crampons is not rocket science. Just make sure you put them on tight, so they can't work their way loose if you try walking sideways across a slope. The ice axe on that route... if you're anywhere that you might slip, it's going to be planted in the snow/ice up above you. If your crampons break loose, you'll need to hang on like crazy. The snow/ice slope will be so rough in June that you won't have much chance to self-arrest. Just make sure you plant the axe and crampons so they won't slip.
I've hauled overnight gear up the MR twice to sleep on the summit. It's fun, but on short trips, you usually have a rough night due to the altitude issues, and can't wait to exit next morning.
So... if you still plan on coming on your trip with the permit uncertainties, you might want to think about having a Plan B, such as hiking in from Horseshoe Meadows, and climbing Mt Langley just a few miles to the south. It's mostly a trail all the way, so a lot less mountaineering, but will surely give you the high-altitude experience. You also need overnight wilderness permits for that, but not nearly as hard to get.
