I know there are many Catholic priests who also belong to the educational sector of the Church, so I just want to know how this is done and how they applied to this job.
A lot of the time they do not apply for the job but are given the job by their Bishop. In my Diocese you'll often find our new young priests at a school for a few years teaching or just being chaplain.
You would find order priests teaching in schools (usually high school) more than you would see diocesan priests. There may be diocesan priests assigned to high schools as chaplains and also do some teaching, but most diocesans have parish duties that would prevent them from being full time teachers. Our parochial vicar goes into the elementary school to teach a religion class here and there, but he is not the religion teacher.
It is an assignment that can be given to some priests by their bishop, as already mentioned. However, the primary way that priests get into the teaching business is by being in a religious order or congregation that concerns itself with this. The Dominicans and the Jesuits are both well known for teaching in universities. I imagine that there are plenty of other religious priests who are involved in education.
My province of Carmelites (Chicago) has 5 High Schools in the United States. Three that we run fully and two that are shared with groups of sisters.
I am currently in my internship phase of my formation for the priesthood and am currently assigned to our High School in Tucson, AZ. The other schools are in Encino, CA, and the other three are in the Chicago area.
But this would all be part of your discernment. As you must also feel called to be a religious. If you do not feel a call to community then you must look to either the diocese or to religious communities (like the Jesuits) that do not stress the common life as much as a mendicant order or monastic order would.
A diocesan priest is assigned as the Chaplain to our Catholic High School. He teaches Sophomore Theology, which is moral ethics (euthanasia, just war, contraception, abortion, etc.) . He does a great job from what I’ve heard.
I believe he also teaches Senior Theology which is Apologetics, but not sure. We have a very strong Theology staff at our high school - all 3 lay teachers have Masters in Theology, so we’re very blessed!