There are three galleries to climb up to in the dome. The first is the Whispering Gallery reached by 257 steps (30 meters high). Do not start the climb if you do not think you can make it as it's one way up and another way down. (The stairway gets too narrow for passing.)
Go to the Whispering Gallery with a friend and stand on opposite sides and face the wall. If you whisper facing the wall the sound of your voice will travel around the curved wall and reach your friend. It does work!
If you choose to continue up, the Stone Gallery offers some great views as it's an outside area around the dome and you can take photos from here. It is 376 steps to the Stone Gallery (53 meters from the cathedral floor).
At the top is the Golden Gallery reached by 528 steps from the cathedral floor.
Find out about more Tall Attractions in London.