October is a very pleasant month to travel around in China. But you'd better take the tour after Oct 07 since the first week of the month is the National Holiday. It will be a little bit crowded in some sightseeing spots. You many check the following page for the climate of Beijing.
http://cdn.travelchinaguide.com/climate/beijing.htm

Generally speaking, 3 or 4 days are enough to visit the highlights in Beijing. If you have ample time, you may plan one or two extra free days to explore or discover the city on your own.