In what contexts would one use the slang word "minging" in British English?

The most common uses are probably to describe people (and to a slightly lesser extent, objects), with the meaning "very ugly" or "repulsive". Less common but still frequently it can mean "smelling very bad".

It is also found frequently as a verb ("That food mings"; "She mings to high heaven"), with either of the senses above, and also commonly with an "-er" suffix to denote a person not possessed of physical beauty ("That psmears is a right minger").

The New Oxford American Dictionary has:

minging (adjective; British; informal): foul-smelling.
• very bad or unpleasant: I'd really like to burn that minging beige jacket he has glued to him all the time.
ORIGIN 1970s: perhaps from Scots dialect ming ‘excrement.’