Actress Billie Piper is not a natural blonde (her natural hair color is a matter of debate), so for the role the answer is yes, and her roots are visible in some episodes, such as "New Earth".
In-universe, the answer is unclear. In "Bad Wolf", Anne Droid implies that Rose uses peroxide in her hair: "So you are unemployed, and yet you still manage to buy peroxide" Rose didn't really get a chance to respond before she had another question thrown at her, and Anne Droid was based upon real-life TV personality Anne Robinson who was known for insulting guests, so whether this was an accurate statement is unclear. On screen, beyond this one event, the subject of her hair colour never came up.
In the expanded media, novels have been contradictory, with one book having her thinking about bleaching her roots, and another having her upset at being accused of bleaching.