Without the use of chemical herbicides, you're going to have weeds. With that said, I wouldn't kill the clover. It doesn't look bad. Just try to mow it enough to keep it from flowering.

As far as seeding, rent an aerator or have a service do it, then seed. By aerating, it will allow good seed to soil contact. After seeding you can put a very light layer of compost over it and water lightly at least once a day until the seedlings are about an inch tall. Then you can scale back on the watering.

Make sure April is the right time to plant. I don't know what your climate is like, but you may need to start earlier if you're planting fescue or bluegrass.

A lot of people like clover. I believe they ignore the bees. My favorite buzzy insect is the paper wasp. I am nurturing a nest of them outside my back door. I like them because they clean up the caterpillars from the yard and, if I don't bother them, they don't bother me.

Would you be willing to spray vinegar on the clover? You could try that on a hot, sunny day (hah, hah, Portland!). Maybe just a sunny day. You only spray the leaves, not the soil. Vinegar should kill the tops. Spray again if/when it comes back. It will come back from seed until the seed is all gone.

Seeding over existing grass is done all the time, but usually in the fall when the underlying lawn has settled into a routine. In the spring you will be fighting against the clover trying to grow fast on the nutrients stored over the winter. If you try it, scalp the grass/weeds as low as you can before you start. You want to get the seed in contact with the underlying soil so the roots will grow into soil. Roots will not grow into air if the soil they sprout into is being held above the soil by other plants.

What kind of grass were you thinking? I think rye is very common in the PNW.

You might be able to keep the weeds out by mowing your grass at the highest setting on your mower. Tall grass shades out the weed seedlings underneath and stops them from photosynthesizing.