What you are talking about is gene therapy.
The simplest explanation is that we don’t know exactly how it would work yet. But we have a pretty good idea of what is required.
As others have mentioned, the easiest way to introduce DNA into a cell would be a virus. Viruses are little DNA-injecting machines, so we have taken certain viruses, gutted their genomes, and can put our piece of DNA in there in place. Viruses aren’t really alive so they don’t really need any genes to survive (although there is size constraints and DNA binding sites in the viral genome that are necessary for correct packing and such). A regular virus turns a cell into a virus-producing factory. A gutted virus would introduce its DNA, we would hope for incorporation, and we would go from there.
Let’s take twoan example. It is not one that you are probably familiar with, but it is one of the more tractable genetic diseases for gene therapy – ornithine transcarbamylase (OTC) deficiency. This disease is from an inborn deficiency of the enzyme ornitihine transcarbamylase. The goal is to reintroduce this enzyme into a tissue that could get rid of the toxic accumulating substances – the liver. So a modified adenovirus, the type that causes a common cold, was produced carrying that gene and not a lot else. The hope was that this virus would infect the liver (it carried proteins that bound liver cells), and introduce the enzyme. These cells would divide and provide enough OTC to get rid of the toxic substances. Unfortunately, one of the test subjects, a 17 year old boy named Jesse Gelsinger, developed hepatic failure and went into ARDS and died. The body responded to the virus in a bad way, and perhaps responded as well by attacking its own liver and this led to multi-system organ failure.
Besides that tragic failure, there are other roadblocks. Take a disease like cystic fibrosis – one would have to target a subpopulation of stem cells in the respiratory epithelium. This is because the respiratory epithelium is constantly regenerating and being shed. So targeting the outer layer of cells won’t cut it, you have to target the few stem cells right on the basement membrane. Unfortunately, these cells are mostly dormant, are only a tiny fraction of the total population, and are usually resistant to any kind of uptake.
There are other ways to introduce new genetic material besides with viruses. One of the more successful is for immune deficiencies and cancer syndrom. Making transgenic cells in a culture dish is a lot easier than doing it in a patient’s bone marrow. So one takes out a bone marrow sample, cultures it, introduces new genes, expands the population, make sure everything is going OK. Then one kills the bone marrow of the patient (with high doses of chemotherapy) and reintroduces the cultured transgenic bone marrow. Unfortunately, a few cases of leukemia have resulted from such an approach. There are also other emerging technologies – gene guns and naked DNA introduction and DNA-coated beads and the like.
It would be next-to-impossible to change the DNA in every cell at the same time. Gene therapy, for now, is just focusing on correcting mutations and broken genes in affected tissues.