I've noticed a couple of times that the Psalms are divided into 5 books, so I did a little searching and found some account of this. The most interesting reason given is that each of the 5 books corresponds to the one of the books of the Pentateuch. This is an old Talmudic interpretation, and it would take some time considering both the Pentateuch and the book of Psalms to determine that completely. According to that scheme, the first 41 Psalms correspond to the book of Genesis. Since I have recently read both of these, I will look for likenesses.
(In case anyone actually reads this, let it be known that I am referring back to the Psalms and thinking this all out as I type it. I don't have an outline and this isn't supposed to be an essay. Just thinking out loud.)
Just looking at the first two Psalms, this view seems difficult to hold. Now these two Psalms are exceptional in other ways. Whereas almost every Psalms in the first book is ascribed to David (1, 2, 10, 33 are the only exception), these two fall outside of that. The content of the first psalm is fairly universal: blessed is the just man, the wicked will perish. Yet the just man is characterized by the fact that "his delight is in the law of the Lord, and on his law he meditates day and night." The law and the study of it is what makes him just, yet as of Genesis, no law has been given.
This reminds me of another interpretation of these two psalms I heard many years ago, namely, that they are a preface to all of the psalms. The Hebrew word for law is torah, which is another name for the Pentateuch itself. Perhaps by saying the blessed man delights in the law of the Lord, the Psalmist is exhorting his audience to delight in the Pentateuch by meditating upon it through the all of the Psalms that follow. Indeed the dichotomy of righteous and wicked, their deeds and their rewards, is fundamental from the very beginning. So that makes some sense of the place of the first psalm.
The second psalm seems unrelated to Genesis for a different reason. "The rulers take counsel together, against the Lord and his anointed." The anointed of the Lord seems to refer to the king here, perhaps David himself, but God has not yet anointed a king in Genesis or in any of the Pentateuch. Certainly the kings and the nations rage against God very early on: this is why he floods the earth, scatters the builders, and rains fire and brimstone on Sodom and Gomorrah. Yet none of those punished acted against the Lord's anointed, unless one consider the only Anointed One, the Christ. "You are my son, today I have begotten you." This psalm is cited throughout the New Testament about the one Lord Jesus Christ, and since He was in the beginning with God who creates all things, it is fitting that a psalm about Him occur early on.
Another surprise: I was looking for a verse I remembered reading, "By the word of the Lord the heavens were made," since this seems to refer to the Divine Word, and found it in psalm 33--another one of the few psalms here not attributed to David. This makes me want to turn to psalm 10...
Not so interesting as I would think. In the Greek numbering, Psalm 10 is joined to Psalm 9 and therefore does not require an additional heading to name its author. So it remains that 1, 2, and 33 are the only Psalms that remain without an author. Perhaps it is worth noting that tradition states that 33 is that age at which Christ died. Perhaps now it is time to look at the rest of the psalms more closely...
I'm not even sure what to look for. Perhaps it would be helpful just to look carefully at the structure of one psalm, so that I can then look for a pattern in the following ones (for they often seem very similar, except for a few choice lines). Let's take psalm 3 since I've been considering 1 and 2.
 A Psalm of David, when he fled from Absalom his son.
 O LORD, how many are my foes!
Many are rising against me;
 many are saying of me,
there is no help for him in God. [Selah]
 But thou, O LORD, art a shield about me,
my glory, and the lifter of my head.
 I cry aloud to the LORD,
and he answers me from his holy hill. [Selah]
 I lie down and sleep;
I wake again, for the LORD sustains me.
 I am not afraid of ten thousands of people
who have set themselves against me round about.
 Arise, O LORD!
Deliver me, O my God!
For thou dost smite all my enemies on the cheek,
thou dost break the teeth of the wicked.
 Deliverance belongs to the LORD;
thy blessing be upon thy people! [Selah]
The major division seems to be between verses 6 and 7. The first 6 verses are all statements, giving an account of what the situation is, and then verse 7 is a plea, "Arise, O Lord! Deliver me, O my God!" and then reasons for doing this, ending with an optative, "thy blessing be upon thy people." As for the division of the first 6 verses, 1-2 are about the foes, 3 is about the Lord, 4-6 is about the one who is helped by the Lord. Is there an order here? Surely it is when one is in trouble that one most often turns to the Lord: so the he tells the Lord about his foes. Then he realizes who he is talking to: "Thou, O Lord, art a shield about me..." and then one can be at peace, knowing that the Lord holds all things in order.
That all being said, what does this have to do with Genesis? Not much as far as I can see. Even the heading says "Of David, when he fled from Absalom his son." That seems like an odd way to start a psalm that is somehow about what occurs in Genesis. Later I will have to read and see how St. Thomas divides the psalms, since he wrote commentaries on a decent chunk of them. Whoa, I just pulled up the proem, and that alone would deserve a whole post. I'm going to end this for now, but I'll post again about the Psalms and about St. Thomas' teaching on them. Here is the St. Thomas:
(Psalm 3 nabbed from http://quod.lib.umich.edu/cgi/r/rsv/rsv-idx?type=DIV1&byte=2154323)