You hypothesis physically implausible - for starters, the speed at which memories are made and recalled is much, much faster than the timescale in which "writing" and "reading" of DNA happens.
While you can encode a lot of data in arbitrary arrangements of a molecule (just as you can encode a lot of data in e.g. arbitrary arrangements of magnetic polarity in a dense material) if you have the right tools to do that, our cells have different tools specialized to do different processing on DNA molecules. There's no evidence to suppose that these changes in DNA are somehow controllable by external factors as opposed to random variations, and there's no evidence to suppose that these changes in DNA are readable in any other way than the usual creation of proteins required for functioning of the cell.
While you can encode a lot of data in arbitrary arrangements of a molecule (just as you can encode a lot of data in e.g. arbitrary arrangements of magnetic polarity in a dense material) if you have the right tools to do that, our cells have different tools specialized to do different processing on DNA molecules. There's no evidence to suppose that these changes in DNA are somehow controllable by external factors as opposed to random variations, and there's no evidence to suppose that these changes in DNA are readable in any other way than the usual creation of proteins required for functioning of the cell.