Glad to hear you’re focusing some of your visit on Crater Lake National Park. It’s perhaps Oregon’s most unique feature and an attraction I routinely recommend.
June still isn’t considered high season for visiting Crater Lake, but your chances will start rapidly improving for seeing more of the park. While there likely will still be some snow on the ground, and lake visibility could be affected by weather, more of the trails will be accessible. Last year, the park’s north entrance and most facilities opened in mid-June, but the year before that, it wasn’t until the end of the month.
Because Crater Lake is at 6,000 feet above sea level, snow has been known to fly in midsummer, and conditions can change in a flash. Be sure to pack a weatherproof jacket and shoes, long pants and a hat and gloves for the time of year you plant to visit. The only way to know what the current conditions are at the park is by checking its website http://www.nps.gov/crla/planyourvisit/current-conditions.htm