In most other discussion I've read on this issue, he's been classified as INFJ. Much of his writings and his biography does protray an INFJ nature.
This is further seen through the many personality-philosophical similarities he shares with many famous INFJ thinkers and writers: Dostoevsky, Buber, Shestov, Berdyaev, etc. I've already posted Shestov's commentary on the similarities between him and Dostoevsky, even referring to Kierkegaard as Dostoevsky's double.
Of course, I will point out that many of his INFJ admirers have criticised him on many accounts. Buber critiqued Kierkegaard's notion of the suspension of the Ethical, and Shestov claimed Kierkegaard didn't go far enough in his radical sense of faith. But then again, this need not relate to type at all; since the INFJ Solzhenitsyn criticised Dostoevsky on many accounts, despite his great admiration for the 19th century figure.
The constant theme of paradox that runs through Kierkegaard's writing is a common one found among INFJs, or rather Ni types in general. This is especially seen in this remark of his:
"The thinker without paradox is like a lover without feelings. The supreme paradox of all thought is the attempt to discover something that thought cannot think."
As for his biography. I'm not in the mood to get into any details at the moment. I do have Garff's 800+ page biography of him, and there's plenty of information about his personal life.