Is there any China tourism board at the entrance of Forbidden City providing group tour? How much?

Yes, there is such kind of service at the Meridian Gate and the Gate of Divine Prowess. If the group has no more than 5 people, the price is Central axis: 1 hour, 200 RMB, Central axis and west route: 1.5 hours, 300RMB, Central axis, west route, and Treasure Gallery: 2.5 hours, 400RMB. An extra 20 yuan per person will be charged for groups of more than 5 people.
I think you can consider to join a tour group, which will make everything going well. The number of the tour group is flexible.
Generally, a tour to it takes about 2-3 h or more.
I suggest you try TravelChinaGuide, an admirable travel agency. I just returned from my Beijing-Shanghai tour served by TravelChinaGuide. It was wonderful.

If needed, you can contact traveltravelchinaguide.com or tourstravelchinaguide.com.

Nice trip!