Yes. Fly Alaska Air to Port Angeles' Fairchild Airport. You will connect through Seattle via Kenmore Air, a partner of Alaska Air, and a wonderful, family owned, small airline. Kenmore will pick you up at Sea-Tac and take you to Boeing Field. From there they will fly you to Port Angeles, a gorgeously scenic 30 minute flight that takes you over the Olympic Mountains and the Straits of San Juan de Fuca.
Rent a car at Fairchild (reserve in advance). Stay at either the Red Lion or the Olympic Lodge in Port Angeles. If your budget is higher, let me know, and I will suggest posher, prettier places.
You can hop the walk on only, 1 hour, Victoria Express, right in downtown PA to take you for a day trip to Victoria. You won't need your car as buses and taxis can take you everywhere, including Butchart Gardens. In fact, Victoria Express has tours to Butchart Gardens set up. June is the perfect time to see these lovely gardens. The peonies should be glorious, and the blue poppies will be in bloom. You can walk downtown Victoria and enjoy the shops, Royal Museum, Chinese restaurants, etc... This is so much easier than Seattle, where you have to take the three hour each way Victoria Clipper.
Please post in the Olympic National Park Forum for suggestions about what to do in the park. This magnificent park is one of the last great wildernesses left in the lower 48 and offers 3 distinct and beautiful ecosystems: alpine, marine, and rain forest. The road to Hurricane Ridge starts right in Port Angeles. If you love natural beauty and hiking, you will be blown away by this park.