How do I repair and clean a loose, greasy washing machine drum?
You probably have a bad drum hub nut. This is a very common failure of a top load washing machine. Look up your model on a few of the any repair parts sites and locate the hub nut. I suggest replacing the all the parts in the process of the replacing the nut. Should be fairly inexpensive for the parts. The procedure usually involves removing the entire machine casing. Make sure you have the machine unplugged.
As far as cleaning, go ahead and use a simple green type product and clean the edges of the inner and outer tub.
If it is the transmission area, then you have a bigger problem. But I bet it is more likely the hub nut.