Based on your short description, this could also be the 1907 Jack London's novel Before Adam. You can read the full English original text on Wikisource, or the same on Project Gutenberg.
This novel tells the story of a first-person protagonist who has vivid dreams in which he experiences the memories of a particular ancestor who lived a few million years ago.
A long science-fiction explanation of this is given in chapter 2. This explanation mentions evolution, and clearly claims that the protagonist has inherited the memories through his line of ancestry. The book does not mention DNA by name specifically, instead it blames “what Weismann terms the ‘germplasm’”, but that's not surprising, for molecular genetics wasn't discovered until a few decades after this novel was published. I will quote some of the explanation below.
[From Chapter 1.] And further, these dream trees were not a mere blur on my vision. They were sharp and distinct. I was on terms of practised intimacy with them. I saw every branch and twig; I saw and knew every different leaf.
[From chapter 2.] It was not till I was a young man, at college, that I got any clew to the significance of my dreams, and to the cause of them. […] But at college I discovered evolution and psychology, and learned the explanation of various strange mental states and experiences. For instance, there was the falling-through-space dream--the commonest dream experience, one practically known, by first-hand experience, to all men.
This, my professor told me, was a racial memory. It dated back to our remote ancestors who lived in trees. With them, being tree-dwellers, the liability of falling was an ever-present menace. Many lost their lives that way; all of them experienced terrible falls, saving themselves by clutching branches as they fell toward the ground.
Now a terrible fall, averted in such fashion, was productive of shock. Such shock was productive of molecular changes in the cerebral cells. These molecular changes were transmitted to the cerebral cells of progeny, became, in short, racial memories. Thus, when you and I, asleep or dozing off to sleep, fall through space and awake to sickening consciousness just before we strike, we are merely remembering what happened to our arboreal ancestors, and which has been stamped by cerebral changes into the heredity of the race.
[…] It will be noted, in passing, that in this falling dream which is so familiar to you and me and all of us, we never strike bottom. To strike bottom would be destruction. Those of our arboreal ancestors who struck bottom died forthwith. True, the shock of their fall was communicated to the cerebral cells, but they died immediately, before they could have progeny. You and I are descended from those that did not strike bottom; that is why you and I, in our dreams, never strike bottom.
[…] yet it is not myself that I see but one that is only remotely a part of me, as my father and my grandfather are parts of me less remote. This other-self of mine is an ancestor, a progenitor of my progenitors in the early line of my race, himself the progeny of a line that long before his time developed fingers and toes and climbed up into the trees.
[…] I possess the memories of one particular and far-removed progenitor.
However, the part of your question about “is extended further and further into prehistory” doesn't match.