How would it be received if I asked during a job interview what my chances are?

Whilst it is understandable that you would like to know this, this question is nevertheless inappropriate in a job interview.
In most cases, the recruiter won’t be able to say anything about it at this stage, as there are likely to be other candidates still under consideration. Furthermore, you risk coming across as uncertain and casting doubt on your suitability. It is therefore better to ask about the next steps in the selection process, or when you can expect a response.