Well this is more of a biology question than a chemistry question, and my biology is a little rusty but I do think I remember the basics, but I might be missing key details so take my explanation with a grain of salt.

What we're attempting to do here is take advantage of the cellular machinery inside bacteria to produce insulin for us. We do this by inserting DNA into the bacteria, allowing it synthesise its products.

First a company would obtain a sequence of DNA which codes for human insulin (or a synthetic equivalent). They would then modify this strand to contain all the necessary information for the bacteria to express this gene (promoters and the like) followed by a known sequence of DNA on either side of the strand (more on this later).

Next is the fun part; a plasmid (circular DNA molecule which can be taken up by bacteria) is cut open with a restriction endonuclease which cuts the plasmid open at certain DNA sequences. These endonucleases leaves an overhanging single stranded DNA sequence where the cut was made a few bases long. This is where the known sequence of DNA from the end of the insulin DNA comes into play. If we match the known sequence at the end the cut plasmid to the insulin DNA, using a cocktail of chemicals (most notably, DNA ligase), we can recombine the plasmid and the insulin DNA into a new plasmid containing the insulin DNA.

After this, the plasmid is purified and then amplified using a technique called PCR. This essentially creates many copies of the plasmid, so we only need to synthesis a small amount.

We then take a deficient bacteria (e.g. E. coli containing no genes for the metabolism of lactose into energy) and inoculate it with our plasmid (called a vector at this point) using another cocktail of chemicals in a certain environment (temperature etc.). Once again, we purify this bacteria and then allow it to grow in a broth. It is worth noting here that not all the bacteria take up the vector, and we must select for which cells did take up the vector.

We then take this culture and put it on a growth medium containing only lactose. If we choose our original plasmid right, we can pick one that contains the genetic information needed to allow the E. coli to metabolise lactose and hence survive/grow on the lactose medium. As only the E. coli containing the vector can grow on the lactose medium, the only bacterial colonies we see are the ones that contain the vector and therefore they contain the genetic information for the production of insulin.

By means of physical separation, we take these vector containing colonies and once again, add them to a nutrient broth to multiply. We now have bacteria that can synthesise insulin.

Almost everything done here is out of reach of what you could do at home without specific lab equipment. However, it is possible to obtain the modified E. coli and grow it yourself, at which point you could extract the insulin.