Choose a couple of SATB (soprano, alto, tenor and bass) vocal music sheets. Select a variety of vocal pieces to use as audition material; this way, you can audition for singers with the ability to handle different vocal styles. You can purchase SATB music at most music stores or at any number of online sheet music outlets. You'll want to find singers who are fluent reading music, so using actual vocal sheets for your auditions will allow you to judge reading ability.
Place an ad in the paper advertising for experienced vocalists with the vocal types you need. A typical vocal quartet will have a soprano, an alto, a tenor and a bass singer. You can also approach members of a church or school choir about forming a vocal quartet. You don't need to have the same setup. It will depend on the type of vocal music you'll be performing, but having a vocalist for each of these parts will help maximize the types of vocal arrangements your quartet will be able to perform.
Audition singers individually first. Listening to how a vocalist performs alone should be the first step in the audition process. Have each candidate read and perform his part from the sheet music you've selected. You want to hear solo singing as well as harmonizing. Ideally, each member of the group will be able to perform a vocal solo if necessary.
Audition your candidate with other members of the quartet if you're satisfied with his solo audition. You want to make sure he is harmonically cohesive with other members of the quartet; you'll also look for personality qualities. A vocal quartet needs to be in sync on more than just a musical level. You want the members of your quartet to fit seamlessly, both on a personal and a musical level. It's important to be sure the four of you have the same goals for the quartet.