Theology and DoctrineThe Church of Christ descended largely from Alexander Campbell, a Presbyterian who rejected the historic and confessional understanding of the Bible. Campbell was an Arminian who so strongly rejected his church roots that he decided to split off altogether and form a new denomination, but fearing the guilt he faced, decided to not call it a denomination.
Almost 200 years later the fruit of their decision is evident, without the creeds to anchor the church and with every believer left to understand things for themselves, there is no unity or common purpose in the group.
The good news is that the tattered remains of their thoughtful Presbyterians persist to some extent. The liturgy is similar, they are Coventalist (as opposed to dispensationalist) in nature, (if you didn't understand that the general rule is that they believe the book of Revelation is past), and have simple singing (so simple in fact it's a-capella). Their leadership is run by a board of elders. If you are a church of Christer unhappy and wondering where to go outside of this group, make for the Orthodox Presbyterian Church and return to your roots.
The bad news is that although they are believers they believe in utter incompetence of every other denominations understanding of the Bible. Their theology is not lifegiving, it's bitter, proud, arrogant, rote, joyless, mechanical, simple. Their faith is small, their boasting of their works great, their numbers collapsing, their studying dreadfully weak, in discipline unloving, in most things dreadfully, willfully ignorant and argumentative. But they are by in large real loving Christians who are captured by a dark wordview. Or perhaps I should say are members of a cult, trapped.
It's exactly what would happen if you reject your heritage and all it's greatest thinkers. They have been cast adrift having no solid foundation for understanding the Bible.
What really sets the CoC person apart and unique is their belief in the pivotal, all surpassing importance of credo-Baptism, communion at every church gathering, the rejection of all creeds, statements of faith, confessions and chatechisms, and their embrace of a-capella worship. Protestants, whatever else you do don't describe their position as Baptismal Regeneration- you will never understand them if you keep using that kind of language. The concept of being born again has no meaning, because that is predicated on a change in nature, or a change in state. The protestant concept of justification means that a change of state is permanent, with the Church of Christ believer this is decidedly not the case.
Church StructureEach church is run by an elder board of at least 2 but no larger than 9 men who have vocations not in ministry. They are responsible for ultimate policy decisions and official church discipline. The church is supported by Deacons, men only, who take care of things, classroom staffing, communion passing, money counting, budget etc. Men may become Deacons after they have young children (1 Tim 3:12), and Elders are appointed from the Deacons after they have teenage children who believe (in accordance with Titus 1:6).
There are three kinds of Churches of Christ: the mainstream, the out of mainstream, and the antis. Even amongst the same group the churches themselves have little to no affiliation with each other, although some bonding occurs when out of necessity they will bond together, eg: to make a land purchase for a summer camp. The mainstream is typically a larger congregation, more strongly liberal, the out of main is a smaller congregation that has likely split off of a main or an out of main (splitting is very common and happens on average every 5 years), and anti are defined by what they are against (ie: no kitchens in church, no doughnuts or food in church, one cup communion only).
Talking to Them
Accept that they think you are likely stupid or shallow, and have not grasped their points about where you are wrong theologically. This is to be expected, as the only reason they have held together as a denomination is this glue. Sure hubris makes a terrible binding agent, but that's what you're working with.
They are most likely going to be very eager to do a Bible study with you, to correct your understanding of things, that's your opening. As you study (most likely on a topic) go to a passage of scripture and press that clear meaning and that alone. Don't let up and don't let them squirm out of it, but instead show them there can only be one meaning and it's the obvious one. Acts 13:48 comes to mind as a good example for this on showing predestination of the elect. You will not make any headway the first few times as they will be trying to convert you, but show a few such passages and press firmly. They will walk away in full disagreement, but pray for them and they won't walk away unchanged. It will likely take 6 to 8 months to fully deprogram them if you minister to them- do so with your whole heart.
They will be very glad you did.
Note: I was very amused when someone found this blog post while searching for "How to deprogram a Church of Christ person." I'm glad there are people out there committed to such a good work. If this is what you are after you are going to need a good block of time for continual study over the coming weeks. If they are willing to study with you just ring the bells of grace in their ears long enough and God will break through. Personally I recommend Romans. Not all of the CoC people are created equal, some who are very unhappy with the churches can be deprogrammed relatively quickly (a few months) but some who really loved the way they operate may take up to a couple years. And, if you can, get them into another church service, to dispel their notion that all other Christians are stupid and unsaved.