PHOENIX — When Kyrsten Sinema began her rise in Arizona politics in the early 2000s, she was a Ralph Nader supporter and local spokeswoman for the Green Party who worked to repeal the death penalty and organized antiwar protests after the Sept. 11 attacks.

But today, as the Democratic nominee for Senate from Arizona in one of the most pivotal races in the country, Ms. Sinema is campaigning as an altogether different person. While she is now a three-term member of Congress, Ms. Sinema is running as much on her biography — her three years spent homeless as a child — as on any issue. She is using that personal hardship to project grit and distinguish herself from “most people in politics,” as she says.

This emphasis on her life story has had a dual effect: It has highlighted her lack of a strong political identity and it has drawn scrutiny to her story of homelessness and some contradictory elements in it.

Ms. Sinema’s evolution reflects a calculation about what it takes to prevail statewide in Arizona, which has not elected a Democrat to the Senate since 1988. And it is a strategy that has put her in a competitive position against Representative Martha McSally, her Republican opponent, as they seek to replace Senator Jeff Flake and Democrats try to upend the current one-seat G.O.P. majority in the Senate.