How does scientists use biochemistry DNA and RNA as evidence of evolution?

In any practical sense, they don't. Scientists have long ago stopped looking for "proof of evolution", since virtually every observation in Biology turned out to be "proof of evolution". So today, even though every new observation and every new experiment still poses a potential test for evolution, the central theses of evolutionary theory are taken as a given.

The ways that observations in biochemistry, especially concerning DNA and RNA, are consistent with common descent and descent with modification are diverse. For one thing, the molecular make-up of RNA and DNA sequences in life forms seemlessly follow the nested hierarchies of life that we find mirrored in morphology, behaviour and fossils. These sequences allow us to formulate more accurate phylogenetic hypotheses. Also, the biochemical details of DNA and RNA and their ancillary mechanisms show us how evolution works at the molecular level.