You can't modify an animal's genes. You can modify the genes for later generations. The easiest way is cross breeding. Breed animals of the same species with different traits together, and you will get an animal with modified genes from it's parents.
Well I reckon that many of the techniques that Monsanto uses on plants can be applied to animals. Just hire a bunch of geneticists and DNA experts and hack yourself a genetically modified creature.
You can splice certain genes into certain organisms. I did a lab back in high school in which we injected a certain gene into plasmids of Escheria Coli and made them glow. It has also been done in goldfish. I'm not sure yet if they've done it on animals, and I doubt there is an easy way to do if it is safe.
You can modify an animal’s genes, including humans,such as in gene therapy. The first gene therapy on a human occurred in 1990, as of 2014,some 2000 other clinical trials have been conducted. For example treatment of haemophilia, treatment of chronic lymphocytic leukemiam, treatment of acute lymphocytic leukemia ec cetera. The most common method uses DNA that encodes a functional therapeutic gene to replace a mutated gene. The gene is packaged within a vector which carries the gene inside the cells. A vector is a DNA molecule used as a vehicle to artificially carry foreign genetic material such as genes into another cell, where it can be replicated and expressed.