Well, what you're looking for is an IVR combined with a billing application. CCM doesn't come with an IVR, so you have to buy one, and then there's the matter of compiling billing records and reading them to the user (so you need TTS licenses). I'm not sure if such an application already exists, you should check the cisco partner catalogue for that.
However, this has nothing to do with the chosen telephony solution.. you can have the same for an Avaya, Nortel or Alcatel system, and it does not make use of any specific Cisco features. So, I think what would be more useful to you is an XML service that allows people to get billing information to their screen. That means no IVR and no TTS (so the price is much lower), and it can be done rather easily (look at the CDR definition and developer documentation, then combine this with an XML service).
Two things to consider: 1) it should not be possible for person A to get the records for person B so you might want to protect access to that service by a password, that in turn will tell you which user is logging in and then you present the billing info for just that user. If you do not do that, you might very well run into serious privacy issues, that could lead to a lawsuit against your company.
