There are a few trips I can think of - the most obvious fact is that snow and trail conditions will be the overriding consideration. We had a very snowy winter and a rather wet spring. The snowpack is very heavy at the moment in the North Cascades. A big hot spell would change things certainly, but it is tough to predict what might happen. And if it does happen, there will be blowdown to contend with early in the season in some spots.
That much said, think options. Truthfully, no matter where you need to go, you gotta get to the trailhead. I wouldn't necessarily restrict myself to one or two trips.
Let's start with a loop in Hannegan Pass area to Whatcom Pass and back via Copper Ridge. You can make it a loop, you can do it a couple of ways. It's about 52 miles with 13,600 feet of gain. Typically done in 5 to 6 days, but a lightweight hiker can go faster. The thing that will likely hold you up is snow on the trail or mud early in the season
Alternative, you could the hike from Hannegan Pass via Copper Ridge all the way over Whatcom Pass to Ross Lake. It's about a 54 mile go of it. Of course, when you get to the end, you are going be a long car ride back to Bellingham.
From Stevens Pass to the town of Steheiken along the PCT you can hike 105 miles through the Glacier Peak Wilderness and along the flanks of its namesake volcanic peak. THis is a challenging piece of trail, with some mighty ups and downs. BUt it is also very beautiful and honestly, generally rather uncrowded. You can get to Steheiken and take a ferry down to the other end of the lake to Chelan, where you would need a way to get back. (You can also continue along the PCT north if you would like to - Steheiken is a TINY town (just a few places) but it is on a lake, is charming and is one of the best trail stops on the PCT (the bakery is fantastic).
Another great loop is one that is often referred to hear - the Spider Gap to Lyman Lakes to Cloudy Pass to Image Lake to Buck Creek Pass. It's mostly a loop (there is a road walk at the end and it's best to find a hitch those last few miles). It's a classic Cascades loop, beautiful and fun. Takes, oh, I'd say four days but you can easily stretch that with some rather worthwhile side hikes. This is a favorite, although honestly, it isn't crowded. Oh, Image Lake might be, because of its iconic stature, but really, it's great to share it with a few people. Miner Ridge is worth exploring. It's about 53 miles with the side hikes around 11,000 feet of gain.
If the North Cascades proves to be snowed in that time of year, don't despair. There are many fine loops in the Olympics. I would recommend the Enchanted Valley - LaCrosse Basin loop in Olympic National Park. If you take a few side trips this is around 66 miles and 13,000 feet of climbing. Beautiful lakes, a great ridge walk, and very likely, a number of bear sightings in the Enchanted Valley itself. The Olympics will meltout before the North Cascades because of the elevation differences.
Great books that I think will help you plan your trip:
Backpacking Washington by Douglas Lrain (Wilderness Press)
Hiking from Here to WOW: North Cascades by Kathy and Craig Copelan (Wow Guides) (Formerly the Don't Waste Your Time in the North Cascades. Regardless, very opinionated guide which I find very helpful).
Some other websites:
Northewest Hikers http://www.nwhikers.net/
Awesome bunch of people who will give you the scoop on trail conditions and anything you ask of them. Great message boards. Excellent trip reports.
Washington Trails Association http://www.wta.org
Another great website full of trip reports. Highly recommended. Great organization that really does right by the trails. If you live in Washington, there is every reason to join.
Finally, besides checking on trail conditions I'd call the ranger districts for the most up-to-date information. Have fun on your trip!