The sole source of a Christian's peace and hope is in Christ and His justifying work on behalf of the sinner. Finney did not believe that Christ's death was anything more than an example of how much God hated sin.
Consider this quote from his Systematic Theology Lecture on Justification -
[i]"That the penitent soul remains justified no longer than this full-hearted consecration continues…"[/i]
Here he is saying that a man's justification is completely based upon his consecration. This is not Biblical as there are multiple verses that tell us a man is justified by faith alone, and that faith will work.
Here is another quote-
[i]"Our own works, or obedience to the law or to the gospel, are not the ground or foundation of our justification … These are conditions of our justification… None of these must be omitted upon pain of eternal damnation."[/i]
So then what he is saying is that Christ's work is really worth nothing, and it cannot actually save anyone unless they have perfect obedience every day until the day they die. This completely makes salvation hinge upon man and his work, rather than on the perfect One who has imputed His righteousness to us. If there is no imputation, we have no salvation.
The next statements alone should be enough to make anyone who understands the work of Christ upon the cross question this man. Speaking on justification again, he says-
[i]"It is not founded in Christ's literally suffering the exact penalty of the law for them, and in this sense literally purchasing their justification and eternal salvation."[/i]
[i]"Neither is the atonement nor anything in the mediatorial work of Christ, the foundation of our justification, in the sense of the source, moving or procuring cause."[/i]
Now compare this to the Bible-
Isa 53:4 Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him stricken, smitten of God, and afflicted.
Isa 53:5 But [b]he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed.[/b]
Isa 53:10 Yet it pleased the LORD to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the LORD shall prosper in his hand.
Isa 53:11 He shall see of the travail of his soul, and shall be satisfied: [b]by his knowledge shall my righteous servant justify many; for he shall bear their iniquities.[/b]
Heb 9:28 So [b]Christ was once offered to bear the sins of many;[/b] and unto them that look for him shall he appear the second time without sin unto salvation.
There are more, but I hope this helps.