MMM is right, Japan isn't really known for cheap shopping.
But what's your definition of "cheap"?
If simply comparing prices in Japan and Europe, any European-based Japanese designer (eg.
Yohji Yamamoto,
Comme des Garcons,
Tsumori Chisato, etc) would be MUCH cheaper in Japan simply because they are locally made. A sales associate for Yohji told me that prices are twice as expensive in Paris, and three times as expensive in New York. Similarly, Japan-originated brands like
BAPE and
Undercover are also cheaper in Japan. So if you're into those styles/brands, I would definitely stock up, although they are by no means "cheap" even in Japan. And somehow I don't think that's what you meant by "cheap shopping" so I should stop babbling.
So again depending on what you call "cheap," I'd say a one-piece dress or nice shirt for around 5000-10000 yen, or a graphic-tee for 2500 yen is considered normal these days. If that's a manageable price range for you, you'd find good shopping places sprinkled all around. Specifically, shops like
United Arrows and
Beams (chain stores, you'd find them all over) have good selections.
If you're going to Tokyo,
Shibuya 109 is a popular shopping destination for young women but they have a pretty distinctive style that not everyone likes, and prices vary. You could find some bargains there though.
Takeshita St. at Harajuku also has both expensive and affordable clothing that are worth checking out.
You can also check out
Uniqlo (esp. their design-t-shirt-only store unique to Harajuku--great selection and last I went prices were slightly over 2000 yen a piece), I don't know how the prices compare to the UK and French Uniqlos, but I'm guessing they'd be cheaper in Japan.
In a few weeks, you might also luck out with summer sales at department stores like
Parco and
Seibu and the above mentioned Shibuya 109.
If you like vintage clothing,
Shimokitazawa is famous for small boutiques and used-clothing stores.
Oops sorry for the long post....but I love shopping.....
But again it all depends what kind of clothing and budget you're looking for. If you can elaborate, I could try to point you to some more specific places.