There are no Transport for London bus services which run immediately in front of Buckingham Palace. Your best chance of seeing it from a bus would be from the top deck of a service running along Grosvenor Place.
See the map here:
http://www.tfl.gov.uk/assets/downloads/centlon d.pdf
Use the zoom function to enlarge the area around Green Park. The grey roads are bus routes. (Buses don't run along the white roads).
As you'll see, the following buses run along Grosvenor Place (but I don't know which ones are double-deckers):
2: West Norwood to Marble Arch
8: Victoria to Bow Church
16: Victoria to Cricklewood
36: Queen's Park Station to New Cross Gate
38: Victoria to Clapton
52: Victoria to Willesden
73: Victoria to Seven Sisters
82: North Finchley to Victoria
148: Shepherds Bush to Camberwell
436: Paddington to Lewisham
Remember that if you've got a Travelcard (for both bus and tube travel) or a bus pass you can use TfL services on a hop-on, hop-off basis. A single bus journey normally costs �2 but a one-day bus pass, for unlimited travel, is only �3.50. A one-day Travelcard, for zones 1 & 2, costs �6.60 (or �5.10 if you don't require travel before 0930 on a weekday).
Chris