It would take as little as 500,000 years with Von Neumann probes. A long time for us but minute in cosmic history...
But still a long time for us, and since WE are the ones taking this undertaking, it is OUR timeframe that needs to be considered here.
DNA: In fact, DNA is a hardy information storing medium, lasting millions of years as well as being hardy when stored.
And if we were talking about exporting our DNA to parts unknown, that would be one thing. But we are discussing practical exploration and/or colonization. Exporting stored human DNA and cloning a fresh population at the destination is neither.
OTOH, the moon, Mars and the asteroids could be settled for relatively little expense, in a fraction of the time and for a fraction of the expense as an interstellar voyage. If you're going to use Von Neuman machines for anything, THAT is the way to go.