What's the main logical or doctrinal basis that is put forth by those Christians who don't believe in the existence of God? I mean, in the Bible Jesus spoke of God the Father many times, so how do they logically explain their belief that God doesn't exist?
As an alternative to "Christian Atheism" as a belief system, there is another interpretation here of a Christian who does not believe in God, which is the "cultural Christian". This was reminded to me by a comment on the question referring to "Jewish" as both the adherents and / or the society - but is has been identified that this is very genuinely the case for Christianity too. This obviously is not based on a scriptural position, but on society and observation; there really are a large number of people who will identify (when asked in passing) as Christian, by which they mean (typically) they have been raised within Christianity, but don't really believe in it as such (and don't typically attend a church or worship, except maybe things like "saying grace" at mealtimes through habit and expectation).
This may be tied to the fact that to say "no, this is not what I believe, and I shall separate myself from it" can sometimes take a lot of effort and energy, jeopardise friendships and relationships (and humans are social creatures) and prospects (perhaps impacting business relationships). It can also cause friction within a family, etc. I can fully accept that it may be a lot easier to just "go along with", since that takes no effort (not even church attendance etc) - especially if their stance one way or the other on the matter is not of huge importance for them (compared with, say, more pressing needs like feeding the family and paying the bills). There are also sometimes additional factors if they want to get their kids into a particular school (alarmingly common), where rocking the boat may cause complications.
So - looked at from that perspective, the cultural Christian (in reality, not a true believer) is a very real phenomenon. This is not, however, a Biblical basis - it is, simply, an observation on the complexities of
(removed "modern", as there is no reason to suppose this is in any way new)
Based on an example of proponents of Christian Atheism, the logic behind a 'Christian' who does not believe in God/god, seems to be:
"I can call myself Christian if I strive to be like the ideal (and not real) concept of Christ. I do not have to believe in the divinity or divine origin of Jesus or Christ."
I don't know that there's a "main" (in the sense of "universal and accepted by all atheist Christians") one, but I can certainly speak for myself:
The assertion that "[believing] in the divinity and/or divine origin of Jesus Christ" is the "definition of 'Christian'" is not one I accept. It is my position that the Christ is an ideal, and that it is a mistake to identify the Christ with Jesus personally. Jesus was but an imperfect manifestation of the Christ. Thus, we do not necessarily need to accept everything Jesus said. Much of what he did points us in the general direction of the Christ, but it remains to us to separate the wheat from the chaff and continue fleshing out our understanding of the Christ and Christness. Jesus's importance comes not from his supposed literal divinity, but as the originator of the tradition of Christ-seeking in which I consider myself to be working within. While many of those Christ-seekers, I believe, made grievous errors, the ultimate goal is still the same.
This Christ is an ideal, an example we should strive for, because by living in this ideal manner as exemplified by the Christ all people can live among themselves without destroying one another (thus, the Christ is very literally our savior, by showing us how to avoid our own destruction).