What is The Black Flame that Anton lavey mentions in his books,The Temple Of Set mentions in there writings,and other Left Hand Path organizations writings and how do you cultivate The Black Flame according to The Church of Satan, The Temple Of Set,and other Left hand path organizations?

It is that unique energy pattern, the spark of vital existence that burns within the Satanist.

You cultivate it through living your life at its fullest.

Since this board is for and by The Church of Satan, you'll have to contact the other organizations for their input.

However, since all other organizations are weak copies at best of the originator of all of this, ie Dr. LaVey and The Church of Satan, you need not look further.