Chemical consideration of biodegradeability
Biodegradability is not a chemical property, much like a price for which you can buy a compound is not its chemical property. Both are the properties caused by some external circumstances, which are not necessarily related to chemistry.

As for the processes, there is hardly anything they have in common. Life is pretty versatile and can consume a wide range of chemical compounds. Wide as it is, though, this range does not constitute all chemistry. There are a great deal of compounds that just never happened to appear in nature before humans made them, so no life form had the chance to develop the ability to consume them. Those are not biodegradable.