There are of course several ways to approach carriers in your efforts to get appointed. However, there is one good approach that is often overlooked.
Assuming you have developed some extent of a network of agency friends and business acquaintances, find out which ones already represent the carrier or carriers that you have determined will be best for your needs. Ask them if they will refer you, in terms of a simple call by them to their carrier rep or someone in a higher position, if feasible. Beforehand, give the potential 'fellow agent referrer' a list of your references, along with key strategies you will employ to attract the kind of profitable business that you have determined the particular carrier likes to write. You should know this based on your prior research about the carrier.