The answer is rather simple. First of all, to be called a Doctor of the Church, the saint must have shown an extraordinary degree of holiness. Now, if you have read my perfection series, you would have learned that not all saints are equal, nor are all people. We have all been called to a certain level of holiness.
Those who have been given the title of Doctor of the Church reveals great holiness. One only needs to think of SS. Bernard of Clairvaux and Therese of Lisieux as examples.
The second criteria is that the person must reveal a high degree of learning, through writings, such as sermons or treatises, or other books. SS. Thomas Aquinas and Teresa of Avila give us good examples, as do all the Doctors, such as Augustine, Ambrose, and today's saint, Peter Chrysologus.
The third category is rather simple. The Church, through the authority of the Pope, must declare the saint a Doctor of the Church. For example, The Pope Emeitus declared two saints as Doctors on October 7th, 2012: SS John of Ávila Hildegard of Bingen.
St. John Paul II declared St. Therese, the Little Flower, as a Doctor of the Church in 1997. Doctors are not declared very often.