Hebrews' allegory of Christ as Priest
The writer of Hebrews presents an allegory that sees Christ achieving what the sacrificial system would have ideally liked to do but couldn't. The new covenant, as a result of Christ, has not only accomplished everything the old covenant did, but surpassed it in every way. I think such an allegory is not really the best place to be looking to learn how the new covenant actually works. But reservations aside, since I have been asked what I think the writer of Hebrews thinks about the atonement...
The first atonement motif in Hebrews is in chapter 2 and talks about him defeating the devil and freeing us from the fear of death, though this motif does not seem to reoccur. So I'm going to put it to one side.
The most common atonement motif is of Christ as an intercessor who submits prayers and supplications to God like an old testament priest. It is in this context that it is said Christ as priest submits to God "gifts and sacrifices" (5:1, 8:3, 9:9) along with his supplications. The idea of sacrifices as gifts to God in order to please him and thus have him view the petitions and prayers favorably was extremely common in the ancient world. The pleasant smell of incense and the giving of gifts other than animals were understood to function in much the same way, thus aiding the petitions of the worshipers.
In this sense in trying to "work out" the allegory, we could say that the writer sees Christ's life of obedience to God and his faithfulness to death doing God's will forms a pleasing gift to God, a sacrificial offering of himself and his life to the will of God ("I have come to do your will" 10:7-9), thus pleasing God and strengthening his petitions.
A second and different function of sacrifices is discussed later in Hebrews - that of purification. In the ancient world the life-force of of pure animals was considered to be able to purify that which it touched by virtue of its own purity. Thus comments are made about how things are washed clean and purified through blood, as if blood was a detergent. The Mosaic law contains long descriptions of how and where to spread the blood to achieve purification. Furthermore the eating of the holy meat of the sacrifice was also believed to bring purification.
Christ gets paralleled to this notion of sacrificial purification. He has achieved some sort of purification that is better and bigger than what the sacrifices could achieve. But how it works is somehow different - we neither are literally washed in Christ's blood nor eat his flesh (well there's the Eucharist, but it's not mentioned). The mechanics of how he achieved this purification are not specified unambiguously or clearly. It is therefore possible to link any atonement mechanism into this.
The mention of a defeat of Satan right near the start of Hebrews would suggest that Christus Victor or Ransom from Satan might provide the correct mechanism for understanding the author's thought here. On the other hand, what is occurring here is sacrificial supersession - ie the sacrificial system is being superseded and replaced by something else. Since cultural studies have indicated that sacrificial systems seem to be always superseded in cultures by morality and ethics this would seem to suggest an Ethical or Moralistic mechanism of atonement is most likely in mind. Throughout the rest of the new testament there is clear evidence of moral living as superseding sacrifices. The most famous being of course Rom 12:1 "present your bodies as a living sacrifice, holy and acceptable to God, which is your spiritual worship", but there are heaps of others.
It would be passing strange if the author of Hebrews saw CV or Ransom as providing the mechanism of supersession when the rest of the NT Christians thought moral transformation to living a moral life in obedience to the teachings of Christ and the imitation of his life was what brought about purification and hence superceded the sacrificial system. These ideas seem to make sense of Hebrews when applied to it. The author of Hebrews also seems to have a tendency to draw moral conclusions from sacrificial sentences, implying the same idea of morality as achieving purification and thus having done away with the need for sacrifices.