It takes some technology that most people don't have (like a centrifuge) but the actual process is simple and can be achieved by anyone with familiarity with the equipment itself.
This could be a case for using shotgun / whole genome sequencing - we'd expect to see a more even distribution of genome coverage than one would get by leaving targeted oligos behind. But this isn't likely to happen anytime soon; the costs are far too high (10-100X greater than targeted sequencing).
In the meantime though, one highly feasible avenue for spotting synthetic DNA oligo fragments is the presence and position of nucleosomes (https://en.wikipedia.org/wiki/Nucleosome), which are DNA-associated protein complexes that occur in DNA from organisms, but not in synthetic DNA oligos. These have been used successfully to trace tissue of origin in cell free DNA in humans, and have specific signals related to chromatin/genome topological state. The only way I can think of to fake this signal would be to have a cell culture from the person you're trying to imitate. Granted, this too is not completely unreasonable - there are now very robust commercially available protocols for deriving iPSCs from small dermal fibroblast samples.
It's scary to think prosecutors have been pitching DNA as infallible, damning evidence all along. It means getting a defense attorney off their game might very well lead to an incorrect conviction (think Serial, though obviously the circumstances there are more tied to the use of the cell phone records). I don't think that technology is going to bring us easier convictions for a while yet (e.g. statistical analysis of evidence leading to inference ala Watson might be interesting). Somehow this isn't reflected in how juries have chosen, though, so I think public education about it (even via entertainment) might be the most effective way to change this.