The background check is usually checking back with your prior employers that you actually did worked there. From my experience, usually the background check happens after you accept the offer. But there typically would be a clause in the contract saying that if you don't pass the background check then the contract would no longer be valid (or similar terms).
If asked, you can say you don't like the firm, but you also need to have a very solid reason of what is the specific thing that you don't like and why. Make sure that the reason you give doesn't backfire as you apply to the new firms (e.g. cannot be you don't like the long hours obviously).
Application aside, I think it would be helpful to do some reflection on your 2 months experience - why you got fired, is there anything you learn from the 2 months, and how can you apply the learning to the next job.