Getting Kerik's phone records so they could be leaked would be no easy task, but I can think of three (pretty much lawful) ways right off the bat:
(1) a subpoena by the plaintiff in the civil suit, on the phone company.  
(2)  Through a public records request.  Depending on whether the phone he used was a state gov't issued phone or not and upon NYS law, the records of his calls may have been public records.  I recall a big interpartisan fight a number of years ago in a northeastern state, in which the out-of-power party alleged that the in-power folks were using their gov't issued cellphones to conduct political business, and sought the call records under that state's open public records law.  While the court came down somewhere in the middle on access (recognizing the inherent potential for political harassment), in the context of litigation alleging Bernie called his GF (or she him) on a gov't phone and plaintiff suffered for it, I'd see a strong argument in favor of granting the plaintiff's request.  Once out there in discovery, I don't think a protective order would work too well - too juicy.
Similarly, it's pretty newsworthy, and the papers may have done a good job of reporting, themselves, resorting to the public records law.
(3)  A clerk in the office that pays phone bills, made a copy and gave it to someone.  Assuming, again, it was a gov't paid phone.