It's on my (long) list of trails to hike... but don't rush it -- Mt. Rainier is incredible, you're going to want to savor that place.
In October, be ready -- the weather can be all over the place. I've been in a snowstorm there in the end of September, and I've been up there with beautiful weather in November, December, and February (by which time the snow was probably around 40 feet deep in Paradise).
It's most likely going to be wet and chilly, and you might get snow -- call ahead and inquire about the conditions before you go, in case there's enough early snowfall to create some avalanche hazards -- which is unlikely in October, but possible.