You gain DNA automatically while your disease progresses - first by infecting people, then by killing them off. I think it's a set rate dependent on numbers, but there doesn't seem to be any wiki entry backing me up on this either way. This is normally your primary income source during the mid-game, after all countries have been infected, but aside from making sure your disease spreads and kills you can't affect that rate in any way.
Popping bubbles - infection or mutation bubbles - is your secondary source of income, and that one you /can/ affect -- even if it's only by default of letting them expire without popping them.
As you progress, however, you can unlock DNA strands that increase the DNA gain from one of those activities in the Genetic Code subsection:
- ATP Boost gives a bonus of about 9 DNA at the very beginning, which helps get started;
- Cytochrome Surge gives increased DNA gain from popping orange bubbles -- in practice, this seems to be around 1-4 extra DNA per;
- Metabolic Jump gives increased DNA gain from popping red Infection bubbles at around the same rate of Cytochromic Surge;
- Catalytic Switch gives bonus DNA for breaking blue 'cure research' bubbles;
- Metabolic Hijack automatically pops existing bubbles for you.
Of the five I'd personally rate Metabolic Hijack the most useless -- if so many bubbles are popping up you're worried you can't get them all in time you can just pause the game for a few seconds, and Metabolic Jump the most useful one in easy/normal because it can and almost certainly will pop up to 58 times.