What is the best route for a road trip from Seattle to Bishop California?

If by "best" you mean "most direct": I-5, OR-58, US-97, CA-139 and US-395.

HTtY

Fastest? Most scenic?

When -- makes a HUGE difference in several parts of the route..

How many days do you have?

Try viamichelin.com or google for itineraries. It will give you mileage and time, which will allow you to plan your stops.

If I were starting the trip today, knowing how much snow has fallen in the Cascades and Sierras, I would go I-5 all the way to Mt. Shasta CA and spend the night. Take I-5 south from Mt. Shasta to I-80 east just north of Sacramento.
Take I-80 east to Reno and then south from Reno on US 395.
HTtY's route is great in the late spring and summer.
I just got in from blowing 7" of snow off my driveway here in central Oregon.

>>Take I-80 east to Reno and then south from Reno that is one (only one) reason we need to know when this road trip is happening.

Should say >>that is one (only one of several) reasons . . .