Most drugs do not actually change DNA. Instead, many drugs interact with proteins which are made from DNA. Different drugs have different jobs and interact with different molecules. Some drugs for example are inhibitors of certain enzymes. These drugs can be used to compete against normal substrates for the binding site of certain enzymes used in a specific pathway. With the binding of the drug instead of the normal substrate, inactivation of the enzyme can occur and this may block that specific metabolic pathway. This is just an example of one kind of drug.
Although many drugs do not "change" DNA, some may intact and degrade DNA. This is common in cancer treatments which target DNA and break it down in order to destroy and kill the cell. Another way that you may see a drug change DNA is through an indirect activation of a DNA methylase (just an example). By activating a methylase (which is a protein), the drug can cause the methylation of DNA and help control gene expression. There is not a direct interaction between the drug and the DNA. Again, this drug interacts with a PROTEIN (in this case an enzyme) and not actually the DNA itself. Hope this helps