St Paul states very clearly that the individual is responsible for his or her actions, and that makes sense to me. Does this indicate his belief in kharma and reincarnation? And if so the doctrine of reincarnation was part of early Christianity.

Yes certainly, right up to the sixth century. It was expurgated, declared heretical by the Second Council of Constantinople in AD 553. This assembly was in reality the last phase of the violent ten year conflict inaugurated by the edict of the Roman Emperor Justinian in AD543 against the teachings of the Church Father Origen. Justinian had assumed leadership of the Church; imperial edicts, regulated public worship and the Church had to submit for a time to the papacy of the emperor. According to the Catholic Encyclopedia (Vols IV & XI) this Second council of Constantinople was attended by very few bishops and although Pope Vigilus was in Constantinople he refused to attend. Regardless of who made up this Second Council the teaching of the Church Father Origen regarding the nature and destiny of the soul was declared to be heretical and that declaration had far reaching effects. What Origen said, in part, was, “Is it not more in conformity with reason that every soul, for certain mysterious reasons, is introduced into a body and introduced according to its deserts and former actions”. Origen was, of course, by no means the only Church Father who believed in reincarnation; probably most of them did. The doctrine of reincarnation was summarily ousted as incompatible with the weight of prejudice brought against it. Most people preferred to accept the rival doctrine of vicarious atonement, by which their past ‘sins’ were wiped out. Reincarnation indicated an accounting and personal responsibility, while the doctrine of vicarious atonement implied that responsibility and accountability, the debts of wrong doings which had accrued, could be assumed by another. However, the doctrine of reincarnation was never lost to esoteric Christianity, it was always available to those who sought deeply. The great majority of earth peoples believe in the doctrine of rebirth. It is making a comeback to the Christian world in these more enlightened days.