I'm not very well versed on this subject, but I think it would be safe to say that most Nazi persecution of Catholics and/or the Church was of a political nature. The Catholic Church as an ideological powerbase was at times a problem for the Nazi government and as a result attempts were made to constrain their power. And at the end of the day Roman Catholicism is a competing ideology to national-socialism. But, large parts of Germany were and still are Catholic, especially in the south. It would make little sense for the Nazis to go fullblown persecution mode on those people. That would really be shooting itself in the foot. Considering what the Nazis did to certain ethnicities, an individual Catholic had very little to fear. Clergymen...probably depended on their resistance to Nazism in general.

That said, the relationship between the Catholic Church and Nazi Germany is long and complicated, so reading even the rather extensive wikipedia article can help you get going, especially since your question was rather broad and would require further research on your part or someone who has made this their specialty to give an extensive rundown.

Regarding the eradication of the Church entirely, one needs to remember that if the most fanatical Nazis had their way, there probably would only be Nazism and little else. But I very much doubt that any concrete plans to that effect were ever made. For most of the period, they had bigger problems to worry about. But hypothetically, the dismantling of the Catholic Church as a powerbase would not look out of place on a Nazi agenda.

They were extremists after all.