Brute force can be used to crack passwords by trying a large number of passwords. Why does no one try the brute force approach on every single character of the password instead?
Because most operating systems hash passwords once entered (and store them in a database). This means that the entire password is converted into a collection of cryptographic hashes, random-looking strings of characters into which the passwords have been mathematically transformed to prevent them from being misused.
So, once a password is hashed for a well defined hash, there is no way to attack each character. That is circular logic. To attack each character you must know each character. To know each character you must attack each character.
If by cracking you mean breaking a given hash of password(s) then it isn't possible since hash functions are built to have seemingly random outputs, so for example: The SHA-1 hash of 'Password' is:
SHA-1(Password) = 8be3c943b1609fffbfc51aad666d0a04adf83c9d
but change only the first character ('Sassword') and you get:
SHA-1(Sassword) = 1243263916a8d81369235831d5a224bbae87a86a
As others have mentioned, hash functions are made to appear this way just for this purpose; to thwart attempts to 'crack' them.
If by cracking you mean brute forcing a user's login credientials to a site, then it is even more clear why it wouldn't work - you won't get feedback from the site saying "The first letter was correct, but the rest wasn't", etc.