How are cancer and apoptosis related?

Inversely. Apoptosis is programed cell death. This is a natural process in normal cells, but is reduced, or not present in tumor cells. Therefore the tumor cells live longer or do not die. This is one of the reasons why tumors can keep growing.
Apoptosis and cancer. Apoptosis is programmed cell death. Just like everything- there is a begining and there will be an ending. Naturally, all cells have been pogrammed to die over a period of time. When this function is impaired or loss by mutation etc - cells live past its "use-by-date" and is able to replicate and pass on any faulty machinery to its offspring, increasing the likelihood of the cell becoming cancerous.
Not directyl. Some cancers can be caused by cells having a lack of apoptosis or cell death. Treatment of cancer can lead to apopotosis.
Apoptosis is. Also called: "programmed cell death". Cancer cells may lose the genes needed to eliminate growing cells that take up space, but do not function well and do not play nice with neighbors, hogging blood supply, oxygen, nutrition and making an organ malfunction. So, malignanct cells cycle, grow, and crowd when not needed. Response from chemo rt may reduce size, but need to eliminate stem cell to cure.