It's a lot of driving for sure, but there's no reason it has to be miserable.
Here's a route I'd suggest -
https://goo.gl/maps/VboC89Bo6uv . It entails long first and last days, Indianapolis to Lincoln NE, but splits at Lincoln. Westbound it travels through the beautiful and remote-feeling Nebraska sandhills, up to I-90 just before Badlands NP - you MUST do the easy loop drive through the Badlands, then stops in the Black Hills. (I've shown Deadwood but it could also be Rapid City - maybe an evening look at Mt. Rushmore.) Then across northern Wyoming and Montana to Missoula, then into Seattle.
Coming back, head south to Hood River, Oregon, and spend half a day exploring the Columbia River Gorge. In May the many waterfalls will be full of water from melting mountain snow, and the orchards in the Hood River Valley may still be in bloom.
Then across Oregon to Idaho, with the next night in Twin Falls. Then through Utah and southern Wyoming to Cheyenne, then back to Lincoln and back to Indianapolis.
This would give you three or four nights in Seattle and one to spare somewhere along the road. Below I've included 3 nights in Seattle and 2 in Hood River 'cause it's so wonderful in the spring.
Overnight locations -
- Lincoln NE
- Deadwood SD
- Missoula MT
- Seattle
- Seattle
- Seattle
- Hood River OR
- Hood River OR
- Twin Falls ID
- Cheyenne WY
- Lincoln NE
- Indianapolis
Note you could easily reverse this if it meant hitting Seattle on the right days of the week.