The tool waskly is referring to is also known as a
slide hammer. Once you remove the axle shaft you should be able to manually rotate the inner race of the bearing with it still in place in the bore. To ensure that it is indeed bad.
If it is you can adjust the diameter of your slide hammer attachments so that they contact only the
inner race of the bearing. This will destroy the bearing, but will make it less likely for you to nick the bore while you're slamming away.
When driving the new bearing in it may help to let it sit in the freezer for a few hours first to slightly shrink it. And possibly warm up the axle shaft bore with a butane or other similar torch. If you do warm up the axle shaft it doesn't have to be much, and watch out for old gear oil as you don't want it to ignite.
Use either a genuine, or create a, bearing driver to contact only the outer race of the new bearing. Drive it in with smooth even motions along the axis of the axle shaft. Striking the inner race will destroy the bearing if the outer race is being held in place by the bore. PVC pipe pieces can sometimes be used for this, or you can find better makeshift bearing driver's.
EDIT:
Be aware that if you are using brake parts cleaner to clean the axle tubes and rear differential. That heating brake parts cleaner produces toxic fumes which are highly poisonous! DO NOT HEAT BRAKE PARTS CLEANER! Allow it to evaporate completely before working. I'm not a professional, I just saw
this thread and thought I should warn you.
