First you need to find a boat you like that has four openings. Despite good reviews, I would recommend a live-aboard over the land based resorts in Raja. I've been on the Pelagian plus four other live-aboards in Raja. Of the 40 or so boats plying the waters of Raja Ampat, there are less than 10 that would be in the same category as the Pelagian. I'm not a huge Pelagian fan anyway.
If you don't use an agent, you will need to go to each boat's website and see if there is availability. Many only do full charters, but you can contact them and they will let you know if the party that charter holder (often a travel agent or dive shop) has openings. The boats usually help out the charterers in this manner. It will take some work. Every high end boat will likely be full by now for next May. There are no boats in Raja in June due the windy season. If there are, it's still an iffy proposition. You might want to consider booking for April/May of 2014. Some of the land based ops stay open, but again it is iffy that time of year.
High end boats:
1. Arenui (my fav)
2. The Seven Seas.
3. Damai I or II
4. dewi nusantara
5. Amira
6. Waow
7. Pindito (not luxurious, but solid boat)
Again, most have moved from Raja by May. If they are still around in May, they have often begun the transition to Komodo or North Sulawesi, so you need to look at fall or spring of 2014.
Getting there is easy. From WA you can fly from Seattle to Bali on several airlines such as Korean, EVA, China Airlines (I think). From Bali, your boat will make flight arrangements and get you the tickets.
Did I mention I love the Arenui.
I just looked up a trip list by a travel list
2013 Scuba Diving Charters - Liveaboard Adventures
She has a Damai trip in May that starts in Sorong.