Depends on what you want to see. If all you want to do is drive by Big Sur, you can make it up to Seattle, but as soon as you get there you'll have to turn around and head back down without stopping much. It's roughly a 2500 mile trip for 5 days, that's 500 miles a day, so 8-10 hours of driving each day. And if you plan on taking PCH up to Big Sur from Indio, that's a whole day drive (maybe more) just by itself. I wouldn't recommend it unless you really like sitting in your car.

Instead, I'd suggest you leave Monday, start heading up to Big Sur via the 101 or 1 or any other combinations of freeways to get up there. Take your time and stop wherever you feel like it, get to Big Sur and look around. Then head up to San Francisco for a couple of days. If you like it up there, you can stay until friday, then make a mad dash to LA along the I-5 (6 hour drive). There's a lot to see along the way, and the Bay Area is great if you like the outdoors. Might be cold up there at this time of year, but there are a lot of places to camp along the way.