It's debatable how this is framed - and that matters when it comes to examining what exactly one is trying to spread. Just saying DNA isn't precise enough.
"Clearly the meaning of life is nothing else but to keep your DNA alive"
"Killing any human being means killing any of your 50th grade cousins, that's a piece of your DNA destroyed, free points lost, so that's out of the way."
In these two statements you shift from "your DNA" to effectively "any DNA". The problem with that is killing a distantly related individual can also make room for a more closely related individual. We see that play out, for example, with the tendency of some male animals to kill offspring that aren't directly related to them - so that the mother is then freed up to care for the male's directly-related offspring.
And of course it manifests in various other ways. In fact it seems inevitable that there will always be competition in ways like this - to preserve one's own genetics at the cost of others (whether by directly harming others or just consuming resources so that they don't). After all, resources are always finite. From a small puddle to a galaxy-spanning civilization, there's always a limit to the total number of individuals it can sustain. Consequently, that leads to the argument that life is inherently selfish, even down to the level of individual genes (The Selfish Gene).
So why does this matter? Well, who's DNA are we going to propagate? If we truly say we don't care, then why not pick the shortest DNA from the simplest single-celled organism we can find? (It probably won't be a particularly robust organism, but that wasn't really specified as a requirement).
But then, why pick the intact DNA strand of an entire organism at all? Why not a smaller segment of DNA? You could make a factory that just churns out sludge made up of the shortest segments possible incorporating each of the base pairs. It's still "DNA". It just doesn't do anything useful.
Of course if our solution of "winning the game of life" involves focusing our efforts on something that's doesn't qualify as alive then perhaps we've mistaken what it means to "win".
So lets re-frame it a little and say that winning means propagating some from(s) of Earth-based life as far as possible so that some version of life survives are long as possible. That keeps us at the level of things that are "alive" and focuses us towards things that are robust at survival and self-replication.
Thus what you'll generally want to look toward are bacteria / single-celled organisms, though in some cases larger-but-still-microscopic organisms might be good candidates (such as maybe Tardigrades). Larger more-complex organisms have a tendency to cause trouble for each other. (Just look at extinctions and such that humans have caused.)
Taking our selection of robust microscopic organisms, the next thing to do is spread them as much as possible. Not on Earth, but to every habitable plant we can get to.
The first step towards doing such could be sending probes to every star we can see. Each probe would carry canisters of dormant versions of our selected organisms. Upon getting near its destination star, the probe would try to detect all the potentially-habitable planets in orbit and then maneuver to drop its canisters on them. Ideally the organisms then spread over the planet, pushing toward the goal of having as much life as possible out there.
The next step could be the creation of machines capable of producing such probes as well as reproducing themselves - all with the prime directive of propagating life. So each would travel out to new planets, while bringing along life to colonize the planet. And then set to work producing more self-propagating machines, each of which scoop up some life to take along to new planets thus continuing the cycle indefinitely.
In theory this would continue on until the heat-death of the universe. So maybe you keep some sort of higher intelligence around to research if physics ever gives a way to avoid that end. But now we're very far into the future and speculation about what's possible.
One interesting thing to consider is what happens if our life runs into other life that independently arose elsewhere. Going back to the idea of limited resources, there will be competition between us and them. In this game of life is our DNA valued over their DNA-equivalent? It's basically just taking the question of who's DNA/life gets preserved to a higher level.
