I can't quite get the hang of this. I've been in a few maps where a pro was just demolishing everything like this. I'm asking for specific instructions with an xbox 360 controller.
To circle strafe with a helicopter in any game you have to understand some things about helicopter controls.
There are three main controls, the cyclic, the collective, and the pedals.
To circle strafe (or orbit while facing the target) you will need to tilt the helicopter to the right with the cyclic, while using the pedals to keep the helicopter pointed at what your circling.
To orbit to the right you would tilt the cyclic right, while turning the pedals left. In human terms, you walk leaning to the right(cyclic), and then keep your torso turned to toward what you are circling (pedals).
I'll try to translate to controller controls later.
Update for Comments: If the bird just spins like a top and falls to the ground then you are using to much pedal, and need to tap it less often. More likely you are loosing altitude as you strafe pointing at the target. This is normal helicopter behavior; I'll explain.
When a helicopter tilts in a direction, some of the force used to lift the helicopter is taken away to be used to shift the helicopter. This means you also have to increase the collective, or make the helicopter go slightly up while you execute the maneuver.
In most games there is a balance point of strafe and lift. This means you tilt(cyclic) the bird enough that full throttle (collective) keeps you at the same altitude. After that it's just a matter of rotating the bird (pedals) to keep it pointed where you want it.
Sorry, I haven't been able to translate to xbox controls yet.
This video seems to have some good tips, and shows the controller actions and results on screen. It's PS3, but hopefully the controls are pretty similar, since it's mostly stick action.
The chopper right/left movement controls are relative to the direction you're facing. So the simplest way to circle-strafe with it is to keep tapping one of the controls lightly while you rotate your view slightly to keep the desired target locked.
So, for example, if you want to strafe counter-clockwise around something, center it in your targeting reticule. Then tap the key to strafe right ('D' by default) while using the mouse to keep the target in the reticule. Keep repeating those little adjustments and you'll circle around the target.
I confess I'm not very good at it, but that is how it's done.