In British English (informal), as an idiom it denotes “achieving more success” or “becoming more successful than before”. In other words, if someone or something is on the up and up, it means he/ she/ it is making very good progress and doing well.

Example: the new business line is on the up and up.”

In American English (informal, North American), it describes someone or something as open, sincere, straightforward, honest, legal, etc.

Example: the new management wants to keep every aspect of its work on the up and up, without being involved in anything dishonest or fraudulent.