Summary: They do not wash their clothes since there is not enough water to do that. They use the same clothes with which they came to the ISS. They get their new clothes with the resupply vessels, and their old clothes are burned.
This article explains:
... clothes don't get dirty as quickly on the Space Station as they do on Earth. Astronauts on the Station are living in a controlled environment, so the temperature stays at a constant, comfortable level. And when everything around you is virtually weightless, you don't have to exert yourself physically the same way you do in the gravity on Earth's surface. However, astronauts do have to spend a substantial amount of time each day exercising so that their bodies don't atrophy in microgravity, so they do still get a workout.
ISS Expedition Six Science Officer Don Pettit wrote that he changes his underwear once every 3 or 4 days.
In an interview in February, Pettit said that he was still wearing the same pair of shorts he had been wearing since he first arrived on the Station - in November! Even though they have more shorts to change into.
They get their clothes with the resupply vessel, and the dirty clothes are burned together with the garbage:
The Russian Space Agency’s Progress resupply vessels are non-reusable, meaning they only conduct one-way trips to the ISS and then burn up upon reentry into Earth’s atmosphere. Before they depart the station, they're loaded with trash—including astronauts' clothes.
To clean or treat the clothes, several options have been thought of:
Cleaning with a plastic bag:
Expedition Six Commander Ken Bowersox also has a favorite pair he chooses to wear frequently. Even though there's no laundry facility on the Station, Bowersox even figured out a way to wash his shorts using a plastic bag.
He washes the clothes in a bag with a bit of water and soap. See this video (last video of this page).
Grow some tomato and basil plants, Don Pettit once folded an old pair of underwear into a sphere to use as a planter. The idea was remarkably resourceful, considering the lack of soil aboard the ISS.
Russian scientists have also tried using special kinds of bacteria to literally digest cotton and paper underwear. Theoretically, the methane produced by the related biochemical reactions could also be harnessed to power spacecraft. But the offbeat idea was never actually put into use, since it could take up to a decade to determine the right combination of bacteria for the job.
NASA's Intravehicular Activity Clothing Study:
The Intravehicular Activity Clothing Study (IVA Clothing Study) dresses crew members in commercially available lightweight clothes that have been designed to resist odors.