It's hard to talk specifically about consciousness, given that we still lack a widely-accepted definition of what it even is, as far as I know. That aside, the question can be rephrased to a more general form: assuming pure materialism (i.e. consciousness arises from some material interactions in the brain/neocortex), how can complex high-level phenomena arise from DNA information alone?
A short answer is that they don't. Assuming pure materialism, consciousness obviously can arise - you and me are conscious, and we have the required biological foundation to exhibit consciousness. So how can this be? DNA in the form of a genome represents only a fraction of biological information. How genes - which represent a subset of DNA - are actually expressed - i.e. what chemical products end up in a cell based on the genetic sequence - is a completely different matter. As you hint, proteins (e.g. transcription factors) absolutely play an important role. To simplify a bit, each cell in your body contains essentially the same DNA, yet cells exist in different forms (muscle cells, nerve cells etc.) and make up different tissues. An analogy: you may have a DVD disc and know exactly how it is encoded, but without an actual DVD player, this information alone means little. DVD player in this example is to DVD disc what a cell is to DNA. You can have identical DNA sequences and obtain completely different results (i.e. different proteins being produced, which are the main agent of biological activity) depending on what genes are active (i.e. being expressed), when they are active and how they are active. In turn, one gene being expressed can lead to shutting down another gene's expression, or even it's own - the process is not time-invariant. An example of a phenomenon where different genes are active at different times is a circadian rhythm.
In general, gene expression and regulation depend on various epigenetic factors, such as the position of nucleosomes (1). Epigenetics essentially refers to our environment being able to influence our gene expression without actually altering the DNA sequence itself; see (2), (3) for an overview. Indeed some studies in mice show that epigenetic changes acquired by parents can be passed to offspring, although the debate on the role of epigenetic inheritance is still not finished (4), (5).
In short, DNA information does not equal total biological information. My answer here of course is very general and can also be used to answer e.g. "How can something as complex as a human arm be encoded in DNA alone?", but I believe it's still relevant given the details and assumptions of the question.