How quickly does melanoma grow?

It varies. What matters is how deep a given melanoma grows. Some tend to grow horizontally and do not spread. Others, stay small, but grow deep, and will spread quite quickly. If you have any concerns about a given spot, get it checked out.
Don't Wait. There are actually a number melanoma subtypes. Some can grow slowly over many years and some can progress rapidly over months. Melanoma can be completely cured if it diagnosed and treated in the earliest stages. While certain clinical features and someday genetic tests may provide clues, no one should 'play chicken' with a potential melanoma. In all cases, early diagnosis and treatment is the key.
Varies. Different melanoma grow at different speeds. It is very hard to predict their growth rate.