Historically, late July to mid August have been the best times to be out in the ocean on our coast and during that time frame one often would get 3-4 days out of 5 of fairly flat water. The past two years have been different though with more winds from the N, more often than in the previous 10. Getting from Astoria to Westport isn't that tough but one does have to time the trip to cross two bars under reasonable conditions. Westport to LaPush is a longer crossing and there's no place in between to hide. La Push to Neah Bay is a relatively quick trip and after that the rest is fairly easy - especially going eastbound.
The Strait of Juan de Fuca can get nasty but in the summer, it's usually calm winds in the early AM with rising winds mid morning through the afternoon. Often the ocean is nicer than the strait in the afternoon since the strait tends to funnel winds.
The proposed trip is very doable (safely) in a C-Dory 22 or 23. That said, there are several precautions to note. First, the ocean can be quite nasty both in the open and at the bars. The entrance to the Columbia river and Westport can be quite dangerous on strong outgoing tides and are often restricted by the coast guard. Restricted means it's illegal to cross unless your boat meets the specifications of the restrictions. Restrictions can range from no boats under 16 to no boats of any kind may cross. The entrance to the Strait of Juan de Fuca can also be rough on strong outgoing tides but it's a pussy cat relative to the Columbia River bar.
Second, this is not a trip to be done be someone with little or no time in boat or little or no time in the ocean. When I say it's easily doable, I say that with the experience of a few 100 days in the ocean in a C-Dory 22, a C-Dory Tomcat and a few other boats. I've also crossed the Columbia River and Westport bars dozens of times. I now know what the ocean will look like with given forecasted sea states and I know what the bars are likely to look like under various combinations of sea state and tide exchanges. That experience allows me to make better go/no-go decisions and allows me to better handle the boat in various sea states. There's no replacement for experience but there are ways to learn more quickly - going out with others who are more experienced, taking various courses etc. But until you've seen and experienced various ocean conditions in person, it's hard to look at a current or forecasted sea state and know if that will be comfortable, uncomfortable or downright dangerous. For example, 8-10' swells. That sounds dangerous to many. BUT if they are 8-10' long period (20s) swells with little to no wind on top, they are very gentle and easy to manage. 8-10' swells at 8s with 15kts of wind are dangerous to boats our size. Even 3-4' waves can be very difficult to manage if they are short period and mixed with wind. Once you seen and experienced a variety of sea states, you can produce a fairly accurate mental picture of what it's like to be in a given measured or forecasted state.
Third, you need to understand the forecasts and gain some experience with them to understand how reliable/unreliable they can be. If there's a forecasted multi-day window of low winds and calm seas, the middle days of that forecast have a much higher probability of being correct than the beginning or end days of the forecast. Forecasts often shift a day or two and they can be wrong. Forecasts of ocean conditions are fairly reliable 3-4 days out but no so reliable beyond 4 days. For a multi-day trip, you need to leave plenty of flexibility in the schedule (e.g. almost no schedule) to allow for the possibility of sitting in port for several days until the weather is better. Having a backup bailout plan in which someone can drive the trailer to your location is not a bad idea either. The worst decisions often get made when someone "Has to be" at place x by time y and the ocean isn't in a good state.
Fourth, you need to properly equip yourself and the boat for the trip. Good life jackets, good radios (with a backup), good navigation equipment (with a backup), twin engines or a single and a good kicker and radar are all recommended. I have a life raft on my boat but I wouldn't make that an absolute requirement if you are otherwise well equipped and a safe operator. You will be nearshore for all of trip. The waters off our coast are often foggy so I think radar is essential. It's not enough to just own the equipment, you also need to be able to use it well. That's not difficult but again experience is valuable.
While it is faster and cheaper to trailer the boat to the San Juans, the scenery off our coast is also spectacular in it's own way and is very worth seeing. The rocky coastline between La Push and Neah Bay is particularly spectacular (but you might not see it due to fog in a single trip). You're also more likely to see massive numbers of humpback whales in the ocean than near the San Juans (but you can see them there also). It's only about 65-70 miles from Astoria to Westport or a 3-5 hour trip depending on conditions. Westport to LaPush is about 80 miles and La Push to Neah Bay is about 50 miles. After that, you have the Strait of Juan de Fuca. There are many ports along the way there and while it can be nasty, I see it as far less dangerous than the ocean segments. Once you're into the Puget sound/San Juans, the water can occasionally be rough in certain places but rarely is it dangerous (unless you run into one of the many well charted rocks). All the longer segments can be done in one day without pushing it. Most days in late July or August are fairly calm - especially in the early AM. May and June are, in general, much rougher than July and August. So to sum up, such a trip is very doable and can be quite enjoyable with the right preparation and experience. It's not a trip for the novice though.