You won’t really know what you’ve got until you pull the old linoleum up and check the subfloor underneath. Once you have that up you will be able to assess how much of the subfloor will need to be removed and replaced.
Hopefully you will remove all or most of it when you replace the subfloor. Bleach is a popular answer to your question, but soap and water will do the job also. If I were doing the job on my own house, I would use bleach and a scrub brush.