If there hasn't been new snow in the last day or two, you'll probably have a boot path. Otherwise, you may have to break trail. Most of the trail will still be on snow. Depending on weather and when you leave you could get anything from postholing to hard, wind blown crust.
The weather will be a crap shooot. You can plan all you want, but you just never know until you see a reliable forecast, and even then it can be completely wrong. But June is as good a time as any to plan a trip to Muir.
Go for it. Enjoy. Know you're route, know the weather and be aware. You'll be fine. Its a fantastic place (unless some dumb ass lights up in the hut).