Um, I think you've made a mistake here.
The West Rim is not part of the National Park, it is on Indian Reservation Land and is also where you'll find the Skywalk.
The bigger thing in your case is that it is at least 3-4 hours away from Williams, and the wrong direction from where you want to go (its back towards Kingman. It is more quiet, because it is quite remote (requiring long drives on rough gravel) and extremely expensive (about $80 per person if you do the skywalk).
Unless you are talking about West Rim Drive within the National Park, which is more commonly called Hermit Road, as it goes to the Hermits Rest Viewpoint and avoids confusion with "GC West". If that's what you are thinking, you can not drive on that road in summer, access is only via free shuttle bus, to avoid overcrowding and parking problems.