In order for christianity to have a doctrine it has to be based on reality, which of course it's not. So who gives a fuck what its doctrine is?
How do you argue with people that believe in fairy tales? That's the hard part. It's like arguing with a brick wall.
With Christians you argue the basis for their faith not doctrine. When arguing doctrine, you will lose, its fully vetted and logically consistent. Trust me, it is. With non Christian religions you can point out obvious flaws in doctrine, but not Conservative mainstream Christianity.
You cannot even argue innerrancy because the Christian will show you the Dead Sea scrolls, and many archealogical finds that pre-date the councils that supposedly picked and chose what to leave in the Bible an all Doctrine. Maybe there is a little thing here or there that has yet to be substantiated, but core issues have all been found and supported.
Try approaching it from their belief in fulfilled prophecy within the Bible. The Bible itself says if something is not fulfilled or wrong then the prophet is incorrect. Christianity hinges on the fact that there are multitudes of Prophecies that were written about and then fulfilled later. Find a flaw in one of those and the whole thing falls apart.'
You can try to find a flaw in the science of the Bible, like many have tried to do here. For example saying the earth was a circle vs. a sphere. But those arguments fall flat (pardon the pun). Even though there are no real scientific errors in the Bible, in the end the Bible is not a science book and therefore does not in any way purport to put forth anything scientific - but it is claimed to be a prophetic one.
That's why you can easily tear down the JW's that follow the Watchtower and other supposed Christian teachers that have claimed Armegeddon or the rapture will happen on one date or another. They are false teachers and can be set aside.
You can also attack the historicity of the information, but good luck on that. Archeaology has yet to find something that contradicts the Bible. They have some things not yet found, but nothing that is a direct and provable contradiction. For example, if you found that any of the disciples recanted what they said about Christ, especially about seeing him after his death, then you would have something. But all of them went to a violent death sticking to their claims. The Christian will say that who gets tortured and killed for something they know is a lie? Many will do it for what they think is true, but these guys went to their deaths to support their first hand accounts. That's rare, possibly it has never happened. It was not like the Govt's at the time supported Christianity and would kill their families - the Govt's were trying to get them to recant.
Historicity and prophecy. Find an error in there and you take out the Christian.