Don't do south to north.
You are likely to have headwinds every day and you will hate everyone and everything.
Logistically, it's best to start in June in the north. (Early or late June??)
If you can't do that starting in May in the north will likely be wet.
If it's wet you have SW winds, dry NW winds.

I've only done extended touring southbound in late spring and summer.
I have had a few wet days with SW winds and the winds were relentless.
Same goes for the few short segments I did northbound against the wind.
The wind never gives you a break and it wears on you.

If you must go south to north, I'd suggest an interior route.
Not the Sierra-Cascades because it will still be snowbound in May / early June.
You can start with the S-C up to Tehachapi, but then stay in the Sierra footnills.
The Sierra foothills are lovely in May,

Follow the Sacramento River in northern California,
then some challenging riding in Southern Oregon over the Siskiyous.
Like the Sacramento Valley, the Willamette Valley is lovely in late spring.
You can do a loop around Mount Rainier and come into Tacoma along the Green River Trail.

Or -
You can ride on the eastern side of the Cascades which is drier and sunnier - more remote.
(You do not want to ride on US 97 thru Bend - scenic but killer traffic.)
And then ride thru cherry country in the Yakima Valley and take the Palouse to Cascades into Seattle.

PM me if you're interested in details.