Deep learning techniques have shown their superior performance in dermatologist clinical inspection. Nevertheless, melanoma diagnosis is still a challenging task due to the difficulty of incorporating the useful dermatologist clinical knowledge into the learning process. In this paper, we propose a novel knowledge-aware deep framework that incorporates some clinical knowledge into collaborative learning of two important melanoma diagnosis tasks, i.e., skin lesion segmentation and melanoma recognition. Specifically, to exploit the knowledge of morphological expressions of the lesion region and also the periphery region for melanoma identification, a lesion-based pooling and shape extraction (LPSE) scheme is designed, which transfers the structure information obtained from skin lesion segmentation into melanoma recognition. Meanwhile, to pass the skin lesion diagnosis knowledge from melanoma recognition to skin lesion segmentation, an effective diagnosis guided feature fusion (DGFF) strategy is designed. Moreover, we propose a recursive mutual learning mechanism that further promotes the inter-task cooperation, and thus iteratively improves the joint learning capability of the model for both skin lesion segmentation and melanoma recognition. Experimental results on two publicly available skin lesion datasets show the effectiveness of the proposed method for melanoma analysis.