I do it by hand after using the magic wand tool and selecting and deleting parts, but most times I do it all by hand for no reason.
If it's a one colour background, I usually set the paint bucket on 'clear' and play around with the Tolerance setting until it deletes just enough, or keep clicking the paint bucket tool outside of the image where the background is/was and it will delete a little bit each time. If it deletes too much, go to the Tolerance setting.