When you select the camera and click on the dropdown menu for the Transform Orientation, you can click on the
+ button to add a new custom orientation. This will automatically be named after the selected object, in this case "Camera" (I already did this in the screenshot below).
If you use the "Camera" orientation and select an object, moving it with GZ will move it closer/farther from the camera, GY moves up/down and GX is for left/right movement. tapping the axis key twice switches to moving on global axis instead of custom.
However, if you want to only adjust certain axes and not all to the camera, it might be best to place an empty in the scene which has the rotation you need and use this to create a custom orientation.