How to get to Mutianyu Great Wall on public transportation? We will be near Forbidden City.

You can take a taxi to reach Dongzhimen Long-distance Bus Station where you can find the bus 867 to reach Mutianyu directly at 07:00 and 08:30.
If you cannot catch that bus, you can go to the same bus station to take bus 916 to Huairou and then take a taxi to reach the wall.

If you're lucky you can get a taxi (private car taxi) that will take youn there, wait for 3 hours and bring you back for 800RMB.