FYI, you might consider installing the free add-on KillWhite that does just that: removes white in a picture. This can work well for complex pictures where it might be difficult to select the white using the PSE tools. It is installed in the PSE filters folder. As with the built-in PSE filters, you can apply it to just a selected area of the picture.
Since the white is replaced with transparency, you would have to apply the filter either to a duplicate of the Background layer or else a renamed Background layer.
You must save the result in a file type that supports transparency. PSD, GIF, PNG, TIFF, EPS support transparency. JPG, BMP do not; If you were to save it as JPG or BMP you would end up with white background again.