I honestly don't know much about this kind of thing, but I believe the Bible talks about its own form of "karma", but it's referred to as the "Law of Reciprocity" in some circles. This simply means that "a man will reap the crop he sows"; "if one lives by the sword one will die by the sword"; and "Judge not, lest you be judged in the same likeness and the measure you use will be used against you", etcetera. So, it seems, in a sense, that what one does will come back on them either as a blessing or as a curse.
Now, reincarnation is a different animal altogether. The Bible makes it very clear that every man is appointed to die once and after that...the judgement. Reincarnation is not a Christian principle, but has its roots in Hinduism and other pagan cults. I personally do not think Christians could harbor such a concept in their hearts, for to do so would be tantamount to calling God a liar.