Driving from L.A to Seattle and then to Vancouver in Winter. What kind of road conditions to expect?

No one can tell you, at least not more than a week ahead and maybe not even that far, what the roads or the weather will be. If you take the coast the whole way you might well not encounter any snow at all. If you take the interstate, well, I'll just say that about the worst winter driving I've ever done was in the pass between Medford and Yreka, in early February, and I am from Canada. Mind you, when I came back the same way in the first week of March it was sunny and warm and the snow was all gone.
You also need to be aware that conditions can change very quickly or over a short distance. It may be raining right next to the water, but if you go inland a little or a little higher up, then it's snow.
You may be required to carry chains in winter unless you have winter tires. If you live in LA, you probably don't have them. You might want to check the regulations for Oregon, Washington, and BC, though I can tell you that neither winter tires or chains are required in winter in the Vancouver area.

It seems to me that there would be no problems whatsoever if you drive 4 by 4 truck and carrying chains, then for the sack of precaution just hearing the radio broadcasting of weather conditions where you are going. Yip