Using the ACA Pacific Coast Route to get from Vancouver to Astoria is a viable option. The first half or so will go through the Puget Sound region, which is nice. The second half, from around Shelton, WA to Astoria, is doable but as noted by others is not particularly thrilling.
The other routing option would be to use one of the routes outlined in the "Bicycling The Pacific Coast" book:
The book has two routes: an inland route following the Puget Sound/Hood Canal, and the Peninsula route that goes the long way around the Olympic Peninsula. That area will most likely wet in mid-June, but it's worth it if you take the time to make the side trips into Olympic National Park. (If you don't, I would recommend the Inland Route.)
If you don't feel like biking, you can take Amtrak from Vancouver to Portland and then either take the Point bus to Astoria:
Or you can ride to the coast.