Before answering the question 'how does chemotherapy kill cancer? it is worth quickly discussing the differences and similarities between a cancerous cell and a non-cancerous cell in a human being.
The chief difference between a cancerous cell and a non-cancerous cell is that the cancerous cell has acquired a genetic mutation in its DNA that has allowed the cell to avoid scheduling and carrying out its own death—a process known as apoptosis. Healthy cells die when they are supposed to die; unhealthy or cancerous cells do not. This is the most general definition of cancer one can imagine.
Cells divide in a five part process known as mitosis. These five parts, known as cell cycles, constitute the times when a cell is at its most vulnerable to chemotherapy, because various chemotherapy drugs work by throwing a wrench of sorts into the process. That 'wrench' stops mitosis in its tracks and leads to the death of the cell.
For this reason, fast-growing or aggressive cancers tend to respond the best to chemotherapy because these cancers are made of cells that undergo mitosis frequently, making them vulnerable far more often to chemotherapy than other cells.
Traditional chemotherapy acts in a systemic manner. This means that the drugs are applied to the entire body even if a patient's cancer is found to be in a confined region of the body. Chemotherapy drugs are administered in such a fashion that this can't be prevented with standard chemo drugs—such as through an IV. This helps explain the existence of the many side effects associated with chemotherapy. It kills cancer cells and it kills healthy cells. The cells it will kill will be the fast-growing healthy cells—such as those in the mouth and digestive system, the blood, and the hair follicles, among other places.
Combination chemotherapy, such as the CHOP regimen, are created and tested with the understanding that each of the drugs will attack the cell cycle in a different way, the hope being to kill as many of the cancerous cells as possible.
The development of chemotherapy began several decades ago when the US government began testing chemical after chemical to see if it had any anti-cancer activity. It has evolved now to the point where scientists and researchers are focusing on treatments that are progressively less systemic and decidedly more targeted, so that they preserve more healthy cells and cause fewer side effects. One direction is in immunotherapy, while others include variations on immunotherapy and radiology, such as the Zevalin therapeutic regimen.
However, chemotherapy isn't going anywhere any time soon, as despite the side effects, it still provides many patients with effective anti-cancer care.