How do you use a docking computer on the Xbox One (not PC) version of Elite Dangerous?

Once you've purchased and installed a docking computer, it will automatically activate when needed. For this to happen, the following conditions must be true:

1. The docking computer must be powered.
2. You must have received authorization to dock.
3. Flight Assist must be on.
4. Throttle must be at zero.

Once the above conditions are met, the computer will automatically take over and fly in for you. When the auto-pilot has control, there are three options to override it:

1. Engage the throttle in either direction. Auto-pilot releases control until throttle is back to zero.
2. Turn Flight Assist off. Auto-pilot releases control until Flight Assist is re-enabled.
3. Power down the docking computer. This is not as convenient as the other two options, as it is not a bindable control, so I only use it when I actually want the docking computer disabled for long periods.

After you pass control back to the docking computer, it generally has no problem resuming the docking procedure from wherever you are (assuming you're still meeting the aforementioned prerequisites).

The docking computer has two fairly basic patterns it uses, depending on what sort of station you're docking with:

Space Station
1. Fly until roughly in line with the "mail slot".
2. Point the nose in the approximate direction of the slot and roll so that the green side is at starboard.
3. Fly toward the slot, staying on the green side.
4. When near to the entrance, ensure ship is properly oriented for head-in entry. (Up until this point, I've seen auto-pilot practically bring the ship in sideways!)
5. Once inside, roll the ship so that the assigned platform is "down".
6. Fly diagonally "down" and forward until positioned directly above the platform.
7. Deploy landing gear.
8. Finish landing vertically.

Outpost
1. Fly until positioned directly "above" the assigned platform.
2. Point the nose of the ship directly at the platform.
3. Fly towards platform until just before the trigger for the landing display.
4. Re-orient the ship so that the assigned platform is "down" and the nose faces in the direction of the platform number display. (Up until this point, the auto-pilot doesn't take your final landing orientation into account when flying. So, this might seem to take a lot of roll/pitch adjustment.)
5. Deploy landing gear.
6. Finish landing vertically.