Trying to remove grey background

If it will be printed with white ink, one method is to use the Blend If option in layer Blending Options to hide the darkest tones in the image, as shown in the demo below. The example shows a blue background so that the change is easier to see.

First increase the black point of the Current Layer. When you find the right level that hides the darkest tones but leaves the image still visible, then you can feather the break (soften it) by holding down the Option/Alt key as you drag either side of the black point slider.