I have driven that trip many times. As Bob says, it's a great adventure. Take your time and don't be in a hurry.
I've never done the trip in October or April - but my father-in-law did it several times during those months from New Brunswick. The only problem he ever had was in northern BC coming into Yukon. It was snowing. So he just held up and in a couple of days it was melted and he was on his way.
The weather is unpredictable. It could be great - nice fall temps. It could be cold and blustery. But during those months, it will not last. If you get a storm, it will be melted and gone within a few days.
In the Yukon (which is a heck of a lot further north than you'll be going), we says any snow that falls AFTER Halloween will stay to April. Before and after that, it comes and goes.
Even if there's a bit of snow, they keep the main roads clear. I wouldn't worry about it. Go for it and enjoy the trip. It's a really good ride.
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