I think it is regrettable that Finney came to think of himself as a theologian. He was a gifted evangelist but I suspect his great intellect (rather than God's Spirit) led him to write on theological topics. Finney's teachings are filled with apparent contradictions and imprudent use of terms that have meanings easily misconstrued. He also frequently uses the dialog approach found in many older theologians in which the author first argues the case he disagrees with and then answers it in full. One can easily piece together isolated statements from Finney to make him look like he rejected essential Christian doctrines. The only way to really understand him is to read his works for oneself. I think it is fair to say that his ideas are aberrant at some points but one really should read Finney's works for oneself to see what Finney meant by what he said in some of the more notorious, isolated quotations that are found on Calvinist websites that reject Finney as a total heretic.
It is true that Finney flatly rejected the Calvinist concept of atonement. He did not deny the blood atonement of Christ. He addresses this issue in chapter 21 of his Systematic Theology. I'll give you a few quotes from this chapter to give you a feel for its essence:
"The English word 'atonement' is synonymous with the Hebrew word 'cofer.' This is a noun from the verb 'caufar,' to cover. The 'cofer' or cover was the name of the lid or cover of the ark of the covenant, and constituted what was called the mercy seat. The Greek word rendered atonement means reconciliation to favor, or more strictly, the means or conditions of reconciliation to favor; from 'katalaso,' to 'change or exchange.' The term properly means substitution. An examination of the original words, in the connection in which they stand, will show that the atonement is the governmental substitution of the sufferings of Christ for the punishment of sinners. It is a covering of their sins by his sufferings."
"Without an atonement, the race of man after the fall sustained to the government of God the relation of rebels and outlaws. And before God, as the great executive magistrate of the universe could manifest his benevolence toward them, an atonement must be decided upon and made known, as the reason upon which his favorable treatment of them was conditioned."
"The whole Jewish Scriptures and especially the whole ceremonial dispensation of the Jews attest, most unequivocally, the necessity of an atonement. The New Testament is just as unequivocal in its testimony to the same point."
This chapter contains dozens of Scripture quotations and references. Finney did reject the Calvinist idea of limited atonement; many of his statements considered in isolation sound worse than his intent understood in context; and some of his views were aberrant. Still, I do not think a fair and honest study of his teachings as a whole would lead an objective reader to conclude that Finney rejected the biblical revelation concerning blood atonement or that his beliefs placed him outside of the bounds of Christianity.