Unlike many other interviews, customer service interviews usually focus a lot around behavioral analysis than any other trait as the role involves a lot around the applicant's people skills apart from technical knowhow as required for the role.
When interviewing for a role which focuses around customer-service, it is important to note that the interviewer will be testing the applicant's level of patience, people-skills and empathy.
The following is a non-exhaustive list of questions an interviewer's likely to ask the applicant for a customer-service based role
1) Tell me about a time you went the extra mile to help a customer
One of the most commonly asked questions - the applicant's usually expected to narrate a detailed experience where he or she went beyond the daily requirements of their job to aid a customer facing a difficulty. There's no hard and fast rule about how much can be said or not (Usually more detail is better). The experience could include anything from simply listening to an unhappy customer's endless rant to helping them waive an unorthodox amount of fees or even liaison with differing departments in your firm to answer a long-standing query. By successfully answering this question, the interviewer will most likely confirm that the applicant has a very strong customer-service ethic.
2) How would you deal with an irate customer
In a customer service role it is more than likely that irate and angry customers from time to time will meet you over the phone or even face to face (depending on the job). Dealing with irate and abusive customers isn't everyone's cup of tea and the interviewer will be analysing the job applicant's level of patience. In any customer service role, patience is a very prized commodity and anyone who loses their cool at the slightest level of aggression on part of a customer is not likely to survive in the job for long. When answering this question, the applicant should show and convince the interviewer that they've got the patience to deal with angry customers and they have the empathy to turn a situation around an angry customer around by going that extra mile.
Customers are usually irate because they just want to vent and the applicant must always remember that any harsh remarks are never personally directed at them.
3) How would you prioritize your day and your customer requests
What the interviewer's really asking here is how good are the applicant's time management skills. A customer-service role can be a very busy and fast-paced role and if the role involves relationship management then time management is absolutely key. In a customer service role, there'll always be days when the worker will be swamped by all kinds of requests from different customers (with similar size of importance) and would be left to the employee's skill and sense of priorities to govern how he or she would action those requests. An interviewer analyzing the job-seeker's answer to this question will be looking at the applicant's sense of time-management and importance of recognizing the importance of answering certain clients the firm does business with, first due to value.
Hopefully this short article gives an understanding of what can be expected in a customer service interview - I'm certain there are a lot more but the fundamentals have been outlined above.