There are many ways to damage the DNA of a rapidly proliferating population of cells; oxidative damage is one example—albeit a very broad example not especially specific to cancer cells, the cells we would like to eliminate.
Many non-targeted chemotherapeutic drugs exploit other mechanisms of depriving cancers cells of a high-fidelity genome, therefore reducing their competitive fitness or outright killing them due to a lack of functioning copies of essential genes. For an overview of such classes of drugs, see the Wikipedia page on Chemotherapy.
Alkylating agents covalently modify the DNA (along with protein and RNA) of affected cells, causing misreading errors to accumulate as DNA is replicated. Alkylated DNA bases cause misincorporation of nucleotides on the newly synthesized strand that sometimes escape proofreading, causing mutations to be passed on to all daughter cells. Also, if such survailance systems are working in cancer cells, alkylated DNA can cause cell cycle arrest or even programmed cell death.
Antimetabolites disrupt necessary cellular metabolic functions. One common mechanism is to interfere with any chemical reactions catalyzed by the folate family of vitamins/enzymatic cofactors. Folate is necessary for synthesis of some new nucleotides, which are necessary for replicating the genome. A lack of these nucleotides can cause low-fidelity DNA replication (introducing deleterious mutations into the genomes the cells) and cell cycle arrest, blocking division and proliferation.
Anti-microtubule agents interfere with the dynamic behavior of microtubules, a subcellular structure responsible for properly sorting the chromosomes during cell division; microtubules also contribute to cell shape and tissue-invasive locomotion. Preventing microtubules from functioning normally can lead to chromosomal missegregation during cell division, depriving some daughter cells of whole chromosomes (gene loss) and leading to genome instability, which can cause cells to loose fitness or loose genes necessary for life.
These are some examples of chemotherapeutics that do not rely on broadly oxidative mechanisms for their cell-killing action. The point of chemotherapy is to be more toxic to malignant cells than to the rest of the body's cells. Since malignant cells often divide more often than normal body cells in adults, many chemotherapeutic agents are employed with the idea in mind of being more toxic to rapidly-diving cells and less toxic to slowly diving cells.