Can you force human body to mass produce specific proteins?
Yes, using recombinant DNA technology, one can take engineered DNA which has been put in an expression vector, induce it by one of various mechanisms to enter cells grown in culture (they can be from human, mouse, monkey, plant, bacteria, yeast, you name it), and the cells machinery will take over, transcribing and then translating it into protein. 


  does this not mean we can force cells to produce specific proteins we want? Purely hypothetically, ethics aside.


Absolutely, and this is done on a regular basis - sometimes at industrial scales, as a matter of fact. Examples of well-known recombinant proteins include insulin, the blood clotting factor VIII, and therapeutic monoclonal antibodies.


  I also can't quite understand "what" uses that DNA to create proteins


This is an extremely broad question, but Wikipedia's Introduction to Genetics may help explain some of these ideas using less technical terms.The quick answer is that it is the cell's transcription and translation machinery mentioned above that do the heavy lifting. Very briefly, DNA is transcribed into messenger RNA or mRNA, which is then targeted to the ribosome for translation in protein. Think of the DNA as the main instructions for the shop, the mRNA as the working templates distributed to many workers, the ribosome as the machinery which takes the template and produces the finished goods, the protein.